I think the point here is how to interpret or translate the Arabic word "اِنْشَقَّ" in verse (54:1) which often is translated split or split in two pieces, the people who witnessed this event have reported to have seen it. We don't know what they have exactly seen, but maybe they've seen two different parts of the moon in a way which was strange and not familiar to them. He only conclusion from the reports is that the two parts could be distinguished somehow, so that a non-Scientist such as Abdullah ibn Masu'd described it as:
... one part remained over the mountain, and the other part went beyond the mountain ... (sahih al-Bukhari)
He said, "Be witnesses." Then a Piece of the moon went towards the mountain. (sahih al-Bukhari)
And Anas ibn Malik reported:
The people of Mecca asked Allah's Messenger (ﷺ) to show them a miracle. So he showed them the moon split in two halves between which they saw the Hira' mountain. (sahih al-Bukahri)
So the mountains referred to in this ahadith are the mountains of Mekka not that on the surface of the moon, so for sure nobody could see more of the moon as we are able to see now with naked eyes.
And as al-Qadi 'Iyad said (see wikipedia):
It has not been said of any people on the earth that the moon was observed that night such that it could be stated that it was not split. Even if this had been reported from many different places, so that one would have to exclude the possibility that all agreed upon a lie, yet, we would not accept this as proof to the contrary, for the moon is not seen in the same way by different people... An eclipse is visible in one country but not in the other one; in one place it is total, in the other one only partial.
The fact that something can be seen in a location, but not seen elsewhere is reasonably explained in this quote. Note that al-Qadi 'Iyad isn't a scientist but a scholar of fiqh and hadith sciences, but as you see a rational person!
Among the narrators of those reports two can't have been witnesses, but must have heard of it: ibn 'Abbas and Anas ibn Malik as both at least were little children at the time. And Anas was in Medina and the reports seem to indicate that the event happened before the hijrah.
Now to the linguistic perspective: اِنْشَقَّ can mean split or split in two pieces, but can also mean a superficial fissure which could be related to this picture https://upload.wikimedia.org/wikipedia/en/thumb/c/c0/Splitting_of_the_moon.jpg/330px-Splitting_of_the_moon.jpg
Note that some reports seem to indicate a partial eclipse, so that one part of the moon was dark and the other enlightened. This was reported by at-Tabarni in his al-Mu'jam al-Awsat (see here the Arabic original) on the authority of ibn 'Abbas who spoke about a partial solar eclipse. The disbelievers said according to this report: "Muhammad has hexed the sun".
In an other version quoted by sheikh at-Taher Benachour in his tafsir at-tahrir wa tanwir they said: "Muhammad has hexed the moon" this wording goes ahead with the verses following verse (54:1). The same report can be found in almost all tafsir books I've read so far beside those only quoting a split or witnessing the split (unfortunately qtafsir didn't quote this hadith from tafsir ibn Kathir, maybe because he quoted many more ahadith and this one was among the last and from a rather "secondary" source).
Some scholars even said that this verse shows a future event (saying it would happen on the day of Resurrection). This view was refuted by others as the verb in the verse is in past form, which indicates that this happened. Some say that the fact that the moon can be "split" this way was sign for the disbelievers that something similar could happen to earth etc. too so the sign was a warning for them. So far the classical view.
To me the eclipse explanation seems the most convincing and if we check the Nasa calculation we will find that between 610 and 622 only 5 partial eclipses could have been occurred or evidenced in Mekka and only the first of them was at the evening (before sunset). And this would explain why other cultures didn't pay too much attention at this event, as partial eclipses may occur very often and are nothing special, the special thing here is that it happens at the right time at right place! So the word "اِنْشَقَّ" in this verse is very ambiguous and could mean a lot more than we might conclude from first sight! I'd say it describes what the people saw "a split": something well known which at a certain time looked strange as if it was cut into two pieces!
Some sources suggest an other possibility: A meteorite in this seems to be witnessed elsewhere too (This article has been edited too many times, I've read older articles referencing it).
Other sources suggest something the Nasa holds as a theory too (at least to some extent): That Earth and Moon have been "split" but that would mean that this event has occurred "long" ago!
From modern Tafsir
In his tafsir Tafhim al-Quran abul 'ala' al-Maududi discusses the criticism of this event saying:
The critics raise two kinds of objections against it.
- In the first place, they say it is impossible that a great sphere like the Moon should split asunder into two distinct parts, which should get hundreds of miles apart and then rejoin.
- Secondly, they say if it had so happened, it would be a well-known event in the world and would have found mention in the books of History and Astronomy.
But, in fact, both these objections are flimsy.
- As for the discussion of its possibility, it could perhaps find credibility in the ancient days but on the basis of what man has cane to know in the present day about the structure of the planets, it can be said that it is just possible that a sphere may burst because of its internal volcanic action and its two parts may be thrown far apart by the mighty eruption, and then may rejoin under the magnetic force of their center.
- As for the second objection, it is flimsy because the event had taken place suddenly and lasted a short time only. It was not necessary that at that particular moment the world might be looking at the Moon. There was no explosion either that might have attracted the people's attention; there was no advance information of it that the people might be awaiting its occurrence and looking up at the sky. It could not also be seen everywhere on the earth but only in Arabia and the eastern (ands where the Moon had risen and was visible at that time. The taste and art of writing history also had not yet developed so that the people who might have witnessed it, should have made a record of it and then sane historian might have gathered the evidence and preserved it in some book of History. However, in the Histories of Malabar mention has been made of a native ruler who had witnessed this phenomenon that night. As for the books of Astronomy and Calendars, this event might have been mentioned in them only if the event had affected the movement of the Moon, its orbit and the times of its rising and setting. As no such thing happened, it did not attract the ancient astronomers' attention. The observatories also were not so developed that they might have taken notice of everything happening in the heavens and preserved a record of it.
At least this seems to show that it was witnessed by other cultures! If you can read German also take a look into this "paper" which shows other interpretations and explanations.
Ibn Kathir when quoting this story in his historical book al-Bidya wa-nihaya also gave reasons why it wasn't quoted elsewhere saying that disbelievers either forgot about it, didn't witness it as it was short, at night (this would be hard to explain in case of the eclipse theory, but ibn Kathir and all other classical Quran-Interpreters considered at-Tabarani's book as a secondary source, mabe a mistake?), maybe (somewhat) localized etc. or for the simple and obvious reason that it would mean that they would admit it was true while they deny or reject Islam and the miracles of Muhammad()!
Also note that nobody would be exploring or staring at the moon at that time except for the people of Mekka whom had challenged Muhammad () to do such a thing! So the chances are few that people elsewhere have witnessed such an event.
This article seem to follow a theory of a falling meteorite ...
On the whole the miracle of this event -whatever it was and whatever both Muslims and the Mekkan pagans saw- is that it happened just after the disbelievers asked Muhammad () to give them a sign. Note that nobody pretends that the moon was split and put back again as you said.
Some mysteries from the Nasa reports
Here some attempts to explain the findings and observations from the moon:
The origin of lunar sinuous rilles remains controversial. Among the alternatives proposed are lava channels and lava tubes, but fracture control is decidedly apparent in some places (source)
A spider web of cracks on the crater floor suggested to R. B. Baldwin (1968) that the floor was bowed up in the middle. Later, dark mare lavas flooded low areas in the outer part of the floor and covered the cracks (source)