Non-Muslims use the following verses to prove the "Rule of Abrogation", a rule that will justify, accordingly, the contradiction of the verses in the Qur'an:
مَا نَنسَخْ مِنْ ءَايَةٍ أَوْ نُنسِهَا نَأْتِ بِخَيْرٍ مِّنْهَآ أَوْ مِثْلِهَآ أَلَمْ تَعْلَمْ أَنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ
We do not abrogate a verse or cause it to be forgotten except that We bring forth [one] better than it or similar to it. Do you not know that Allah is over all things competent?
Qur'an
Al-Baqara (The Cow)
2:106
وَإِذَا بَدَّلْنَآ ءَايَةً مَّكَانَ ءَايَةٍ وَٱللَّهُ أَعْلَمُ بِمَا يُنَزِّلُ قَالُوٓا۟ إِنَّمَآ أَنتَ مُفْتَرٍۭ بَلْ أَكْثَرُهُمْ لَا يَعْلَمُونَ
And when We substitute a verse in place of a verse - and Allah is most knowing of what He sends down - they say, "You, [O Muhammad], are but an inventor [of lies]." But most of them do not know.
Qur'an
An-Nahl (The Bee)
16:101
Question
Do these verses imply "changing past revelations if Allah deems it necessary"?
Answer
Assalamu 'Alaikum!Arabic for "Peace be upon you", Muslim greeting. Alhamdulillah!Arabic for "Praise be to God", Muslim common phrase. Thank you very much for this question.

As a Muslim, we believe that the Qur'an in its Arabic form is the literal word of Allah (God). Therefore, we need to set a very high standard for this book. It should impress us or at least prove itself to be the best among any other books. First and foremost, the very basic assumption, is that, there should be no contradiction. Qu'ran is a holy book that is consistent with the following: verses, scientific discovery, mathematical patterns, rhyming of the verses from the very first page to the very last page, artistic calligraphy, linguistic composition, and whatnot. But, don't fall into what I say, see it for yourself, and put it into test as Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. challenges all readers of the Qur'an:

  1. Open Challenge 1: The Whole Qur'an
    قُل لَّئِنِ ٱجْتَمَعَتِ ٱلْإِنسُ وَٱلْجِنُّ عَلَىٰٓ أَن يَأْتُوا۟ بِمِثْلِ هَٰذَا ٱلْقُرْءَانِ لَا يَأْتُونَ بِمِثْلِهِۦ وَلَوْ كَانَ بَعْضُهُمْ
    Say, "If mankind and the jinn gathered in order to produce the like of this Qur'an, they could not produce the like of it, even if they were to each other assistants."
    Qur'an
    Al-Isra (The Night Journey)
    17:88
  2. Open Challenge 2: Ten Chapters
    أَمْ يَقُولُونَ ٱفْتَرَىٰهُ قُلْ فَأْتُوا۟ بِعَشْرِ سُوَرٍ مِّثْلِهِۦ مُفْتَرَيَٰتٍ وَٱدْعُوا۟ مَنِ ٱسْتَطَعْتُم مِّن دُونِ ٱللَّهِ إِن كُنتُمْ صَٰدِقِينَ
    Or do they say, "He invented it"? Say, "Then bring ten surahs like it that have been invented and call upon [for assistance] whomever you can besides Allah, if you should be truthful."
    Qur'an
    Hud (Hud)
    11:13
  3. Open Challenge 3: One Chapter
    وَإِن كُنتُمْ فِى رَيْبٍ مِّمَّا نَزَّلْنَا عَلَىٰ عَبْدِنَا فَأْتُوا۟ بِسُورَةٍ مِّن مِّثْلِهِۦ وَٱدْعُوا۟ شُهَدَآءَكُم مِّن دُونِ ٱللَّهِ إِن كُنتُمْ صَٰدِقِينَ
    And if you are in doubt about what We have sent down upon Our Servant [Muhammad], then produce a surah the like thereof and call upon your witnesses other than Allah, if you should be truthful.
    Qur'an
    Al-Baqara (The Cow)
    2:23

Let's analyze the verses being questioned.

First Verse
مَا نَنسَخْ مِنْ ءَايَةٍ أَوْ نُنسِهَا نَأْتِ بِخَيْرٍ مِّنْهَآ أَوْ مِثْلِهَآ أَلَمْ تَعْلَمْ أَنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ
We do not abrogate a verse or cause it to be forgotten except that We bring forth [one] better than it or similar to it. Do you not know that Allah is over all things competent?
Qur'an
Al-Baqara (The Cow)
2:106
Islam is the religion of Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. from the very beginning. Many non-Muslims claim that Islam is a later invention, but it's not, it is the religion of all time. To prove that, let's consider the core belief of this religion:
  1. One Absolute God. Allah is not a "Muslim God", instead Allah is the almighty God of all human beings. Whoever that one supreme being, that's the Almighty God. Allah simply mean "the one and only true God".
  2. Muslims (followers of Islam) simply mean "one who submits to the will of the Almighty God". Whoever submits all his life to the will of God, he is a Muslim by definition.

Therefore, all prophets send by Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. are considered Muslims by definition. That is because all of them submit to the will of Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific., and all of them teach the absolute oneness of God, from prophet Adam (A.S.) to prophet Moses (A.S.) to prophet Jesus (A.S.) (see John 5:30) and finally to prophet Muhammad (S.A.W.)Ṣalla llāhu ʿalay-hi wa-alehe-wa-sallam, Arabic for "May Allah honour him and grant him peace", Muslim honorific..

The revelations of the scriptures where chronologically given to succeeding prophets, and prior to the revelations of the holy Qur'an, Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. first revealed the Torah to prophet Moses (A.S.). There is no doubt Moses (A.S.) taught the core belief of monotheistic religion, and that is, God is absolute in oneness:

שְׁמַ֖ע יִשְׂרָאֵ֑ל יְהוָ֥ה אֱלֹהֵ֖ינוּ יְהוָ֥ה׀ אֶחָֽד׃
"Hear, O Israel: The LORD our God, the LORD is one.
Bible (ESV)
Deuteronomy
6:4

Years after the revelations of the Torah, Allah revealed the Gospels to the children of Israel through prophet Jesus (A.S.). As a Muslim, we believed Jesus (A.S.) was only send to the people of Israel, this is confirmed by Matthew 15:24 (assuming this is indeed a revelation). Therefore, at this point, if Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. is indeed consistent with His verses, then the foundation of the Gospels must be based on the Torah. Further, if Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. is also full of wisdom, then the Gospels must not be identical to the Torah, but rather build on top of it (the Torah), so as to preserve the consistency of the core message of the religion of Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific.. Indeed, that's exactly what happened, since Jesus (A.S.) used the Torah as his basis for preaching but also include other revelations from Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific..

  1. Jesus (A.S.) acknowledged the Torah (the laws and the prophets):
    μὴ νομίσητε ὅτι ἦλθον καταλῦσαι τὸν νόμον ἢ τοὺς προφήτας· οὐκ ἦλθον καταλῦσαι ἀλλὰ πληρῶσαι. ἀμὴν γὰρ λέγω ὑμῖν, ἕως ἂν παρέλθῃ ὁ οὐρανὸς καὶ ἡ γῆ, ἰῶτα ἓν ἢ μία κεραία οὐ μὴ παρέλθῃ ἀπὸ τοῦ νόμου ἕως ἂν πάντα γένηται. ὃς ἐὰν οὗν λύσῃ μίαν τῶν ἐντολῶν τούτων τῶν ἐλαχίστων καὶ διδάξῃ οὕτως τοὺς ἀνθρώπους, ἐλάχιστος κληθήσεται ἐν τῇ βασιλείᾳ τῶν οὐρανῶν· ὃς δ᾽ ἂν ποιήσῃ καὶ διδάξῃ, οὖτος μέγας κληθήσεται ἐν τῇ βασιλείᾳ τῶν οὐρανῶν.
    "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished. Therefore whoever relaxes one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven.
    Bible (ESV)
    Matthew
    5:17-19
  2. Jesus (A.S.) teaches the absolute oneness of Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific.:
    αὕτη δέ ἐστιν ἡ αἰώνιος ζωή, ἵνα γινώσκωσιν σὲ τὸν μόνον ἀληθινὸν θεὸν καὶ ὃν ἀπέστειλας ἰησοῦν χριστόν.
    And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent.
    Bible (ESV)
    John
    17:3
    Καὶ προσελθὼν εἷς τῶν γραμματέων ἀκούσας αὐτῶν συζητούντων, ἰδὼν ὅτι καλῶς ἀπεκρίθη αὐτοῖς, ἐπηρώτησεν αὐτόν· Ποία ἐστὶν ἐντολὴ πρώτη πάντων; ἀπεκρίθη ὁ Ἰησοῦς ὅτι Πρώτη ἐστίν· Ἄκουε, Ἰσραήλ, κύριος ὁ θεὸς ἡμῶν κύριος εἷς ἐστιν, καὶ ἀγαπήσεις κύριον τὸν θεόν σου ἐξ ὅλης τῆς καρδίας σου καὶ ἐξ ὅλης τῆς ψυχῆς σου καὶ ἐξ ὅλης τῆς διανοίας σου καὶ ἐξ ὅλης τῆς ἰσχύος σου. δευτέρα αὕτη· Ἀγαπήσεις τὸν πλησίον σου ὡς σεαυτόν. μείζων τούτων ἄλλη ἐντολὴ οὐκ ἔστιν. καὶ εἶπεν αὐτῷ ὁ γραμματεύς· Καλῶς, διδάσκαλε, ἐπ’ ἀληθείας εἶπες ὅτι εἷς ἐστιν καὶ οὐκ ἔστιν ἄλλος πλὴν αὐτοῦ· καὶ τὸ ἀγαπᾶν αὐτὸν ἐξ ὅλης τῆς καρδίας καὶ ἐξ ὅλης τῆς συνέσεως καὶ ἐξ ὅλης τῆς ἰσχύος καὶ τὸ ἀγαπᾶν τὸν πλησίον ὡς ἑαυτὸν περισσότερόν ἐστιν πάντων τῶν ὁλοκαυτωμάτων καὶ θυσιῶν. καὶ ὁ Ἰησοῦς ἰδὼν αὐτὸν ὅτι νουνεχῶς ἀπεκρίθη εἶπεν αὐτῷ· Οὐ μακρὰν εἶ ἀπὸ τῆς βασιλείας τοῦ θεοῦ. καὶ οὐδεὶς οὐκέτι ἐτόλμα αὐτὸν ἐπερωτῆσαι.
    And one of the scribes came up and heard them disputing with one another, and seeing that he answered them well, asked him, “Which commandment is the most important of all?” Jesus answered, “The most important is, Hear, O Israel: The Lord our God, the Lord is one. And you shall love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.’ The second is this: You shall love your neighbor as yourself.’ There is no other commandment greater than these.” And the scribe said to him, “You are right, Teacher. You have truly said that he is one, and there is no other besides him. And to love him with all the heart and with all the understanding and with all the strength, and to love one's neighbor as oneself, is much more than all whole burnt offerings and sacrifices.” And when Jesus saw that he answered wisely, he said to him, “You are not far from the kingdom of God.” And after that no one dared to ask him any more questions.
    Bible (ESV)
    Mark
    12:28-34
  3. Jesus (A.S.) submits to the will of God:
    Οὐ δύναμαι ἐγὼ ποιεῖν ἀπ’ ἐμαυτοῦ οὐδέν· καθὼς ἀκούω κρίνω, καὶ ἡ κρίσις ἡ ἐμὴ δικαία ἐστίν, ὅτι οὐ ζητῶ τὸ θέλημα τὸ ἐμὸν ἀλλὰ τὸ θέλημα τοῦ πέμψαντός με.
    I can do nothing on my own. As I hear, I judge, and my judgment is just, because I seek not my own will but the will of him who sent me.
    Bible (ESV)
    John
    5:30

At this point, we can therefore conclude that, the new revelations in the Gospels encompasses the previous revelations as it builds on top of it. If you look at the first verse being questioned, it simply says that, the Torah, which we can think of as "old edition" of the holy book, is not abrogated or cause to be forgotten, except that Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. will bring forth better than it or similar to it, and in this case that would be the Gospels. We can think of the Gospels as "new edition" of the holy book. Again "new edition", in the sense that previous edition are not ignored or abrogated but rather are used as reference since the Gospels and the Torah have similarity but not identical!

Analogously, we can think of the Qur'an as the "latest edition" of the holy books. That is, the Qur'an builds its foundation from the previous revelations, the Torah, the Psalms and the Gospels, as confirmed by the following verses:

إِنَّآ أَوْحَيْنَآ إِلَيْكَ كَمَآ أَوْحَيْنَآ إِلَىٰ نُوحٍ وَٱلنَّبِيِّۦنَ مِنۢ بَعْدِهِۦ وَأَوْحَيْنَآ إِلَىٰٓ إِبْرَٰهِيمَ وَإِسْمَٰعِيلَ وَإِسْحَٰقَ وَيَعْقُوبَ وَٱلْأَسْبَاطِ وَعِيسَىٰ وَأَيُّوبَ وَيُونُسَ وَهَٰرُونَ وَسُلَيْمَٰنَ وَءَاتَيْنَا دَاوُۥدَ زَبُورًا
Indeed, We have revealed to you, [O Muhammad], as We revealed to Noah and the prophets after him. And we revealed to Abraham, Ishmael, Isaac, Jacob, the Descendants, Jesus, Job, Jonah, Aaron, and Solomon, and to David We gave the book [of Psalms].
Qur'an
An-Nisaa (The Women)
4:163
وَقَفَّيْنَا عَلَىٰٓ ءَاثَٰرِهِم بِعِيسَى ٱبْنِ مَرْيَمَ مُصَدِّقًا لِّمَا بَيْنَ يَدَيْهِ مِنَ ٱلتَّوْرَىٰةِ وَءَاتَيْنَٰهُ ٱلْإِنجِيلَ فِيهِ هُدًى وَنُورٌ وَمُصَدِّقًا لِّمَا بَيْنَ يَدَيْهِ مِنَ ٱلتَّوْرَىٰةِ وَهُدًى وَمَوْعِظَةً لِّلْمُتَّقِينَ
And We sent, following in their footsteps, Jesus, the son of Mary, confirming that which came before him in the Torah; and We gave him the Gospel, in which was guidance and light and confirming that which preceded it of the Torah as guidance and instruction for the righteous.
Qur'an
Al-Maidah (The Table)
5:46
Therefore, there is no changing in the words of God. From the very early revelations of Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. to prophet Adam (A.S.) to the latest revelation (the final testament -- Qur'an) to prophet Muhammad (S.A.W.)Ṣalla llāhu ʿalay-hi wa-alehe-wa-sallam, Arabic for "May Allah honour him and grant him peace", Muslim honorific., the core message is intact and consistent across time. That's why Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. said in the Qur'an:
لَهُمُ ٱلْبُشْرَىٰ فِى ٱلْحَيَوٰةِ ٱلدُّنْيَا وَفِى ٱلْءَاخِرَةِ لَا تَبْدِيلَ لِكَلِمَٰتِ ٱللَّهِ ذَٰلِكَ هُوَ ٱلْفَوْزُ ٱلْعَظِيمُ
For them are good tidings in the worldly life and in the Hereafter. No change is there in the words of Allah. That is what is the great attainment.
Qur'an
Yunus (prophet Jonah (A.S.))
10:64
وَٱتْلُ مَآ أُوحِىَ إِلَيْكَ مِن كِتَابِ رَبِّكَ لَا مُبَدِّلَ لِكَلِمَٰتِهِۦ وَلَن تَجِدَ مِن دُونِهِۦ مُلْتَحَدًا
And recite, [O Muhammad], what has been revealed to you of the Book of your Lord. There is no changer of His words, and never will you find in other than Him a refuge.
Qur'an
Al-Kahf (The Cave)
18:27
As a conclusion for this section, the verse being questioned is meant to justify that succeeding revelations are builds on top of previous revelations, so as to preserve consistent and therefore no contradiction across revelations as many non-Muslims would say.

Lastly, as a Muslim, Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. knows best on the true meaning of all the verses He revealed, but we believe in it and there is no contradiction in the Qur'an as Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. said:

أَفَلَا يَتَدَبَّرُونَ ٱلْقُرْءَانَ وَلَوْ كَانَ مِنْ عِندِ غَيْرِ ٱللَّهِ لَوَجَدُوا۟ فِيهِ ٱخْتِلَٰفًا كَثِيرًا
Then do they not reflect upon the Qur'an? If it had been from [any] other than Allah, they would have found within it much contradiction.
Qur'an
An-Nisa (The Women)
4:82

Second Verse
وَإِذَا بَدَّلْنَآ ءَايَةً مَّكَانَ ءَايَةٍ وَٱللَّهُ أَعْلَمُ بِمَا يُنَزِّلُ قَالُوٓا۟ إِنَّمَآ أَنتَ مُفْتَرٍۭ بَلْ أَكْثَرُهُمْ لَا يَعْلَمُونَ
And when We substitute a verse in place of a verse - and Allah is most knowing of what He sends down - they say, "You, [O Muhammad], are but an inventor [of lies]." But most of them do not know.
Qur'an
An-Nahl (The Bee)
16:101

Does the Qur'an substitute the previous revelations? To some extent yes, just like how we think of new edition of any Science book, it is meant to substitute the old edition, but that doesn't mean the new edition teaches totally different concepts, but rather it builds on top of the previous edition. Hence, we can use the previous edition as a reference though it was substituted already by the new edition. It's not a contradiction but it's a fact. So we Muslims use the previous revelations as a reference as well, due to the same reasoning, and that is, it has the same core message anyways, and Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. told us to refer to it to confirm the revelations in the Qur'an:

فَإِن كُنتَ فِى شَكٍّ مِّمَّآ أَنزَلْنَآ إِلَيْكَ فَسْـَٔلِ ٱلَّذِينَ يَقْرَءُونَ ٱلْكِتَٰبَ مِن قَبْلِكَ لَقَدْ جَآءَكَ ٱلْحَقُّ مِن رَّبِّكَ فَلَا تَكُونَنَّ مِنَ ٱلْمُمْتَرِينَ
So if you are in doubt, [O Muhammad], about that which We have revealed to you, then ask those who have been reading the Scripture before you. The truth has certainly come to you from your Lord, so never be among the doubters.
Qur'an
Yunus (Jonas)
10:94
See my discussions for 10:94 (a christian friend questioned this verse as well) here.

Verse 16:101 refer to the time when prophet Muhammad (S.A.W.)Ṣalla llāhu ʿalay-hi wa-alehe-wa-sallam, Arabic for "May Allah honour him and grant him peace", Muslim honorific. told the people about the new revelations. The Jews don't care anyway if new verse are substituted by another verse in the Qur'an. They only care if the Qur'an will substitute their scriptures. That is why they accused Muhammad (S.A.W.)Ṣalla llāhu ʿalay-hi wa-alehe-wa-sallam, Arabic for "May Allah honour him and grant him peace", Muslim honorific., as quoted explicitly in 16:101 above, saying "You, [O Muhammad], are but an inventor [of lies]". This line simply tells us that the above verse is meant to substitute the previous scriptures.

Why the First Verse (2:106) Don't Refer to Qur'anic Verses?
If you insist in believing verse 2:106 as referring to abrogation of the "Qur'anic verses", then here's a great explanation from True Islam website. This is an example of why the original text of the scriptures must be encouraged to have in the holy book. For one main reason, to have a ground truth, because translation miss some of the informations. Notice in my posts, I always include the hebrew text of the old testament, and the greek text of the new testament (despite the Christians don't have it by default in almost all of their bibles). This is one of the reason why Muslims recite and study the Qur'an in its Arabic form. Hence, if you happen to ask a Muslim and he says "you need to understand the Qur'an in its Arabic form", then the following will help you understand why.

مَا نَنسَخْ مِنْ ءَايَةٍ أَوْ نُنسِهَا نَأْتِ بِخَيْرٍ مِّنْهَآ أَوْ مِثْلِهَآ أَلَمْ تَعْلَمْ أَنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ
We do not abrogate a verse or cause it to be forgotten except that We bring forth [one] better than it or similar to it. Do you not know that Allah is over all things competent?
Qur'an
Al-Baqara (The Cow)
2:106
The word that we want to give importance is the word "verse" highlighted both in its original Arabic text and in the Saheeh International translation of the Qur'an. We Muslim refer to the verses in the Qur'an as Aayah (or Aayat), spelled in Arabic as آية‎ in singular, and آيات in plural. This is the same Arabic word used in the above verse which is ءَايَةٍ (aayatin). You can check the wikipedia of this word here.

The meaning of the word Aayah in the Qur'an is not limited to the word "verse". It could mean the following:

  1. Miracles or Signs from God as in:
    وَلَقَدْ ءَاتَيْنَا مُوسَىٰ تِسْعَ ءَايَٰتٍۭ بَيِّنَٰتٍ فَسْـَٔلْ بَنِىٓ إِسْرَٰٓءِيلَ إِذْ جَآءَهُمْ فَقَالَ لَهُۥ فِرْعَوْنُ إِنِّى لَأَظُنُّكَ يَٰمُوسَىٰ مَسْحُورًا
    And We had certainly given Moses nine evident signs, so ask the Children of Israel [about] when he came to them and Pharaoh said to him, "Indeed I think, O Moses, that you are affected by magic."
    Qur'an
    Al-Isra (The Night Journey)
    17:101
    And verily We gave unto Moses nine tokens, clear proofs (of Allah's Sovereignty). Do but ask the Children of Israel how he came unto them, then Pharaoh said unto him: Lo! I deem thee one bewitched, O Moses.
    Pickthall
    Other Translation
    17:101
    To Moses We did give Nine Clear Signs: As the Children of Israel: when he came to them, Pharaoh said to him: "O Moses! I consider thee, indeed, to have been worked upon by sorcery!
    Yusuf Ali
    Other Translation
    17:101
    And certainly We gave Musa nine clear signs; so ask the children of Israel. When he came to them, Firon said to him: Most surely I deem you, O Musa, to be a man deprived of reason.
    Shakir
    Other Translation
    17:101
    To Moses We gave nine illustrious miracles. Ask the Israelites; Moses came to them. The Pharaoh said to him,"Moses, I believe that you are bewitched".
    Muhammad Sarwar
    Other Translation
    17:101
    And indeed We gave to Musa (Moses) nine clear signs. Ask then the Children of Israel, when he came to them, then Fir'aun (Pharaoh) said to him: "O Musa (Moses)! I think you are indeed bewitched."
    Mohsin Khan
    Other Translation
    17:101
    And We gave Moses nine signs, clear signs. Ask the Children of Israel when he came to them, and Pharaoh said to him, 'Moses, I think thou art bewitched.'
    Arberry
    Other Translation
    17:101
    In some translations, the word signs above is translated as miracles. From the above verse, we have the word Aayah in plural, hence we have this spelling, ءَايَٰتٍۭ (aayaatim). One might say, it's not the same with the one in Wikipedia in which the plural of Aayah is آيات. Of course, not exactly but close to it, because the one in Wikipedia is only consonants (no vowels -- these are the small lines, small inverted (x-axis) e, above and below the consonant letters). The Arabic letters are calligraphic, has gender (like Spanish, e.g. library is considered feminine), and has long vowels (that gives the correct pronunciation). These are rules that will guide us on when to use آية‎ (aayat), ءَايَةٍ (aayatin), ءَايَةً (aayatan), and ءَايَٰتٍۭ (aayaatim), all of these are the same. Further, because Arabic letters are calligraphic, the forms does change depending on the preceding and succeeding letters. If you don't believe me, you can check the quran.com specifically here (it has a word to word translation of the 2:106).
  2. Portent, Evidence, Example or Sign for people to take heed from as in:
    وَقَوْمَ نُوحٍ لَّمَّا كَذَّبُوا۟ ٱلرُّسُلَ أَغْرَقْنَٰهُمْ وَجَعَلْنَٰهُمْ لِلنَّاسِ ءَايَةً وَأَعْتَدْنَا لِلظَّٰلِمِينَ عَذَابًا أَلِيمًا
    And the people of Noah - when they denied the messengers, We drowned them, and We made them for mankind a sign. And We have prepared for the wrongdoers a painful punishment.
    Qur'an
    Al-Furqan (The Criterian)
    25:37
    And Noah's folk, when they denied the messengers, We drowned them and made of them a portent for mankind. We have prepared a painful doom for evil-doers.
    Pickthall
    Other Translation
    25:37
    And the people of Noah,- when they rejected the messengers, We drowned them, and We made them as a Sign for mankind; and We have prepared for (all) wrong-doers a grievous Penalty;-
    Yusuf Ali
    Other Translation
    25:37
    And the people of Nuh, when they rejected the messengers, We drowned them, and made them a sign for men, and We have prepared a painful punishment for the unjust;
    Shakir
    Other Translation
    25:37
    We drowned the people of Noah because of their rejection of the Messengers and made them evidence of the Truth for mankind. We have prepared a painful torment for the unjust ones.
    Muhammad Sarwar
    Other Translation
    25:37
    And Nuh's (Noah) people, when they denied the Messengers We drowned them, and We made them as a sign for mankind. And We have prepared a painful torment for the Zalimun (polytheists and wrong-doers, etc).
    Mohsin Khan
    Other Translation
    25:37
    And the people of Noah, when they cried lies to the Messengers, We drowned them, and made them to be a sign to mankind; and We have prepared for the evildoers a painful chastisement.
    Arberry
    Other Translation
    25:37
    Here we have it in singular, hence we have it as ءَايَةً (aayatan).
  3. Sign, Evidence, Token as in:
    قَالَ رَبِّ ٱجْعَل لِّىٓ ءَايَةً قَالَ ءَايَتُكَ أَلَّا تُكَلِّمَ ٱلنَّاسَ ثَلَٰثَ لَيَالٍ سَوِيًّا
    [Zechariah] said, "My Lord, make for me a sign." He said, "Your sign is that you will not speak to the people for three nights, [being] sound."
    Qur'an
    Maryam (Mary - the Mother of Jesus (A.S.))
    19:10
    He said: My Lord! Appoint for me some token. He said: Thy token is that thou, with no bodily defect, shalt not speak unto mankind three nights.
    Pickthall
    Other Translation
    19:10
    (Zakariya) said: "O my Lord! give me a Sign." "Thy Sign," was the answer, "Shall be that thou shalt speak to no man for three nights, although thou art not dumb."
    Yusuf Ali
    Other Translation
    19:10
    He said: My Lord! give me a sign. He said: Your sign is that you will not be able to speak to the people three nights while in sound health.
    Shakir
    Other Translation
    19:10
    Zachariah asked, " Lord, show me evidence (if this is a heavenly news)." The Lord said, "The evidence for it is that you must not speak (to any mortal) though you are in good health for three nights (and days)."
    Muhammad Sarwar
    Other Translation
    19:10
    [Zakariya (Zachariah)] said: "My Lord! Appoint for me a sign." He said: "Your sign is that you shall not speak unto mankind for three nights, though having no bodily defect."
    Mohsin Khan
    Other Translation
    19:10
    He said, 'Lord, appoint to me some sign.' Said He, 'Thy sign is that thou shall not speak to men, though being without fault, three nights.'
    Arberry
    Other Translation
    19:10
    Again, in this example we have the word Ayat in singular form, hence we have the exact same spelling of it in arabic, ءَايَةً (aayatan), as that in 16:101.
  4. Verse, Revelation, or Sign in the Qur'an as in:
    كِتَٰبٌ أَنزَلْنَٰهُ إِلَيْكَ مُبَٰرَكٌ لِّيَدَّبَّرُوٓا۟ ءَايَٰتِهِۦ وَلِيَتَذَكَّرَ أُو۟لُوا۟ ٱلْأَلْبَٰبِ
    [This is] a blessed Book which We have revealed to you, [O Muhammad], that they might reflect upon its verses and that those of understanding would be reminded.
    Qur'an
    Sad (The Letter Sad)
    38:29
    (This is) a Scripture that We have revealed unto thee, full of blessing, that they may ponder its revelations, and that men of understanding may reflect.
    Pickthall
    Other Translation
    38:29
    (Here is) a Book which We have sent down unto thee, full of blessings, that they may mediate on its Signs, and that men of understanding may receive admonition.
    Yusuf Ali
    Other Translation
    38:29
    (It is) a Book We have revealed to you abounding in good that they may ponder over its verses, and that those endowed with understanding may be mindful.
    Shakir
    Other Translation
    38:29
    It is a blessed Book which We have revealed for you so that you will reflect upon its verses and so the people of understanding will take heed.
    Muhammad Sarwar
    Other Translation
    38:29
    (This is) a Book (the Quran) which We have sent down to you, full of blessings that they may ponder over its Verses, and that men of understanding may remember.
    Mohsin Khan
    Other Translation
    38:29
    A Book We have sent down to thee, Blessed, that men possessed of minds may ponder its signs and so remember.
    Arberry
    Other Translation
    38:29
At this point, it should be clear that the word Aayah could mean "sign", "miracle", "evidence", "portent", "token", "example" or "verses" (or "revelations"). In islam, we treat verses as signs from God. However, the meaning of the word Aayah really depends on the context of the verse as we've seen above. For example, in verse 17:101 above, we cannot translate the word Aayah into "verses", since that would translate into:

And We had certainly given Moses nine evident verses(?)...

While verses are considered signs in Islam, in 17:101, the nine evidence presented to Moses (A.S.) was not literally the verses of the Qur'an, but "miracles". Hence it is not appropriate to translate it to "verses" when it's not. The same for 25:37, you cannot translate the word Aayah into "verse", since that would become

And the people of Noah - when they denied the messengers, We drowned them, and We made them for mankind a verse(?)...

That would not make any sense if we translate a sign to a verse in the context of 25:37. If we look at 19:10, it is still not applicable since that would sound as follows:

[Zechariah] said, "My Lord, make for me a verse." He said, "Your verse is that you will not speak to the people for three nights(???), [being] sound.(???)"

Now let's check the verse being questioned,
مَا نَنسَخْ مِنْ ءَايَةٍ أَوْ نُنسِهَا نَأْتِ بِخَيْرٍ مِّنْهَآ أَوْ مِثْلِهَآ أَلَمْ تَعْلَمْ أَنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ
We do not abrogate a verse or cause it to be forgotten except that We bring forth [one] better than it or similar to it. Do you not know that Allah is over all things competent?
Qur'an
Al-Baqara (The Cow)
2:106
If we translate it into "a verse" (excluding other meanings), then that would not make any sense. Why?
  1. Because the words "cause to be forgotten" could not be applicable if the word "Aayah" in this verse meant a verse in the Qur'an. How can a verse in the Qur'an become forgotten? For even if the verse was invalidated by another (as the interpreters falsely claim) it will still be part of the Qur'an and thus could never be forgotten.
  2. The words "We replace it with that which is better than it or similar to it" would be meaningless if the word "Aayah" in this verse meant a Quranic verse, simply because it would make no sense for God to invalidate one verse then replace it with one that is similar to it!
  3. If the word "Aayah" in verse 2:106 meant a miracle, an example, an evidence, a portent, or a sign, then all the words of the verse would make perfect sense. The words "cause to be forgotten" can apply to all other meanings except for Qur'anic verses, and that is what actually happened with the passing of time. The miracles of Moses (A.S.) and Jesus (A.S.) have long been forgotten. We only believe in them because they are mentioned in the Quran.
Similarly, the words "We bring forth [one] better than it or similar to it" are in line with the miracles of God. God indeed replaces one miracle with its equal or with one that is greater than it. Consider the following verse:
وَلَقَدْ أَرْسَلْنَا مُوسَىٰ بِـَٔايَٰتِنَآ إِلَىٰ فِرْعَوْنَ وَمَلَإِي۟هِۦ فَقَالَ إِنِّى رَسُولُ رَبِّ ٱلْعَٰلَمِينَ فَلَمَّا جَآءَهُم بِـَٔايَٰتِنَآ إِذَا هُم مِّنْهَا يَضْحَكُونَ وَمَا نُرِيهِم مِّنْ ءَايَةٍ إِلَّا هِىَ أَكْبَرُ مِنْ أُخْتِهَا وَأَخَذْنَٰهُم بِٱلْعَذَابِ لَعَلَّهُمْ يَرْجِعُونَ
And certainly did We send Moses with Our signs to Pharaoh and his establishment, and he said, "Indeed, I am the messenger of the Lord of the worlds." But when he brought them Our signs, at once they laughed at them. And We showed them not a sign except that it was greater than its sister, and We seized them with affliction that perhaps they might return [to faith].
Qur'an
Al-Baqara (The Cow)
43:46-48

Final Message
In this article, I've shown you the explanation and analyses of the verses being questioned to the best I can. However, as a Muslim, Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. knows the best meaning of these verses. There may be ambiguous verses in the Qur'an, but what is important is that the foundation of our belief must be consistent with all previous scriptures. Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. clearly stated this in the following verse below:
هُوَ ٱلَّذِىٓ أَنزَلَ عَلَيْكَ ٱلْكِتَٰبَ مِنْهُ ءَايَٰتٌ مُّحْكَمَٰتٌ هُنَّ أُمُّ ٱلْكِتَٰبِ وَأُخَرُ مُتَشَٰبِهَٰتٌ فَأَمَّا ٱلَّذِينَ فِى قُلُوبِهِمْ زَيْغٌ فَيَتَّبِعُونَ مَا تَشَٰبَهَ مِنْهُ ٱبْتِغَآءَ ٱلْفِتْنَةِ وَٱبْتِغَآءَ تَأْوِيلِهِۦ وَمَا يَعْلَمُ تَأْوِيلَهُۥٓ إِلَّا ٱللَّهُ وَٱلرَّٰسِخُونَ فِى ٱلْعِلْمِ يَقُولُونَ ءَامَنَّا بِهِۦ كُلٌّ مِّنْ عِندِ رَبِّنَا وَمَا يَذَّكَّرُ إِلَّآ أُو۟لُوا۟ ٱلْأَلْبَٰبِ
It is He who has sent down to you, [O Muhammad], the Book; in it are verses [that are] precise - they are the foundation of the Book - and others unspecific. As for those in whose hearts is deviation [from truth], they will follow that of it which is unspecific, seeking discord and seeking an interpretation [suitable to them]. And no one knows its [true] interpretation except Allah. But those firm in knowledge say, "We believe in it. All [of it] is from our Lord." And no one will be reminded except those of understanding.
Qur'an
Ali 'Imran (The Family of Imran)
3:7
Lastly, all the prophets send by Allah (S.W.T.)Subhanahu wa ta'ala, Arabic for "May He be glorified and exalted", Muslim honorific. where meant to glorify Him, not to spread message where people are taught to worship the prophet:
مَا نَنسَخْ مِنْ ءَايَةٍ أَوْ نُنسِهَا نَأْتِ بِخَيْرٍ مِّنْهَآ أَوْ مِثْلِهَآ أَلَمْ تَعْلَمْ أَنَّ ٱللَّهَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ
It is not for a human [prophet] that Allah should give him the Scripture and authority and prophethood and then he would say to the people, "Be servants to me rather than Allah," but [instead, he would say], "Be pious scholars of the Lord because of what you have taught of the Scripture and because of what you have studied."
Qur'an
Ali 'Imran (The Family of Imran)
3:79
References
  1. A. Muhammad. The lie of "Quranic Abrogation". True Islam, www.quran-islam.com. Retrieved August 17, 2018
  2. The Noble Qur'an, URL: https://quran.com/, Retrieved August 18, 2018.
  3. The Qur'an Arabic Corpus, URL: http://corpus.quran.com/, Retrieved August 19, 2018.
  4. Saif min Suyufillah. Response 67. Contradictions in Qur'an Refutation. URL: https://nocontradictionsinquran.wordpress.com, Retrieved August 20, 2018