Salaam Yaaluwa,
You bring up several interesting points relating to philosophy, linguistics, and anthropology. I understand your discomfort with "logic and rationality" discussions. I put this here because belief is fundamentally a philosophical question. Epistemology deals with the study of the evidence of the external world - seeing and believing. This can provide a useful framework for understanding how people react to the evidence of perception. Philosophy helps us to frame questions and to examine possible responses in an orderly and defensible way.
Muslims have objected to philosophy because it raises questions that are sometimes regarded as blasphemous. I am uncertain why we should be disturbed by this more than Christians are but that is another discussion. Philosophy disturbs people generally - Muslims and others - because its constantly changing points of reference require a high tolerance for ambiguity. Educating people to tolerate ambiguity is not in the interests of dogmatists whose primary goal is to teach their view of the correct position on every possible topic, not to teach methods of inquiry and critical analyses of foundational doctrines - this is as true of political orthodoxies as it is of religious ones and you can see it in American society as the boundaries of public inquiry are increasingly restricted by an emerging, quasi-tyrannical political culture.
The controversy over the splitting of the moon is over the context of the reference: does it refer to an event in the life of the Prophet, upon whom be peace, or to a yet unwitnessed sign of the hour. It is cryptic and cannot be disambiguated by grammar alone. There are strong arguments supporting both positions.
You mention the distinction between a vision and a dream. The Arabic word /ru'iyah/ is a noun that means "vision" and can be used to refer to a particular kind of supposedly prescient dream. These dreams of future things are assumed to be a lesser kind of revelation. There is no doctrinal basis for assuming that such dreams are real, and there are strong arguments against the assumption that we might know the future through dreams. Nevertheless, people continue to believe that they can sometimes see the future in dreams. One possible reference for defense of this argument is in Surat Yusuf, where prescient dreams figure prominently throughout the story, both in the case of Joseph and, I believe, of his father Jacob. The argument neglects that both men were Prophets and so legitimate claimants to prophecy which can include a kind of divinely directed future-sight. The nounm /ru'iyah/ is used in Yusuf (12) at 5, 43, and 100; see also al-Israa 18:60, al-Saafaat 38:105, and al-Fath 48:27.
Words derived from /r'y/ are common in the Quran but commonly refer to eyesight and seeing. One derivation refers to the opinion of others or personal reputation. It was the common word for "to see" in the dialect of Quraysh and so it occurs frequently.
The dream you refer to in the story of Abraham could have been a waking dream, the same for the burning bush, and we know today that such lucid dreams are common. Arabic has a word for dream - /hilm/ - and this is the word that is ordinary used to describe the sleep-state consciousness most of us recall experiencing frequently. There is a broad literature describing supposed interpretation of dreams, both early and modern. Freud and Jung were not the first to realize that these experiences could be significant in some way.
I do not agree that Muslims do not make a distinction between visions and dreams. The statement is far to broad. In the first place and depends on there being clear linguistic categories for each of these two experiences. That means it is culturally bound beyond religion. With regard to Arabic, there are clear linguistic categories for each idea, evidenced by distinct words. Whether individuals confuse these two ideas or not depends on their personal understanding. At best, I think, there may be some confusion about lucid and sleeping dreams, but this confusion is not confined to Arabs or to Muslims. Indeed, most people do not understand this well. Lucid dreaming can be frightening since we assume that in a wakeful state, our perceptions do reflect a socially shared reality. This brings us back to the epistemological conundrum: is this really all a dream? explored in movies like Vanilla Sky, The Matrix series, and, of course, Groundhog Day.
In Surah Yusuf, the Prophet says, "Oh Father, I saw eleven stars and the sun and the moon, I saw them prostrate themselves to me." His father, another Prophet, immediately understands the meaning of this and advises him not to tell his brothers about "what he saw". There is no mention of sleeping here and we have no basis for assuming that God may only "show" a prophet a peculiar understanding of events if he is asleep. In fact, this would violate the principle of divine omnipotence.
Finally, you mention the story of Abraha and his ill advised campaign against Makkah in the Jahiliyyah referred to one short sura:
"Hast thou not seen how thy Lord dealt with the owners of the Elephant?"
This refers to an historical event which occurred at least a generation before the Prophet was born. It is a cultural, historical reference. There is no basis to assume that the Prophet "saw" this in a vision or even as an eyewitness, and no reason for making any such claim. The event was common knowledge at the time. It is as if someone had said to you, "Did you not see how the US behaved in the American-Mexican war (1846-1847)?" If you said "yes", I would not understand that you actually saw Grant crossing the Rio Grande in a vision. If you went on to tell me that you actually saw it, I might call 911.
Of course, neither of us actually "saw" this. It refers to common knowledge and understanding - not to dreams, unless, of course, you want it to.

Salaam,
Omar