Great Wheel—1e kinda, 2e, 3e, 5e
You are correct that the Great Wheel cosmology was used in at least 2e, 3e, and 5e. It was codified, and most thoroughly detailed, in 2e’s Planescape setting.
Prior to Planescape, the cosmology wasn’t named, but nonetheless something sort of like the Great Wheel had been gradually developing and emerging from the various discussions and statements about the planes. Dragon vol. 8, in July 1977, included a write-up and sketch of the planes by Gary Gygax himself, that looks and sounds quite a lot like what would eventually be the Great Wheel.
History of the Great Wheel and older editions
The history of Planescape references the prior editions of D&D that had only the Law–Neutral–Chaos axis via the “War of Law and Chaos” that took place in the distant past, in the earliest years of the multiverse’s history. The Planescape approach is to say that Good and Evil, and their relevant planes, existed back then (which is consistent with the original books, I believe—even without Good and Evil alignments, heavens and hells were still discussed), they just were nascent and relatively unimportant, as everything was swept up on one side or another in the war that started between the Wind Dukes and the Queen of Chaos. The Chaotic position in this war was opposed to even the most basic laws like “causality” and “matter,” which kind of explains how it was a concern that trumped even Good vs. Evil. It’s worth noting that Law won this war, and so even the most Chaotic portions of the Great Wheel are at least a little bit Lawful, in that even the most fluid forms have a certain limited amount of consistency and time progresses in a mostly linear manner.
Planar Development: Demi-or-not-Plane of Shadow
The biggest change within the Great Wheel cosmology is the Plane of Shadow. Originally, in 2e, this was a demiplane, but in 3e it had become a plane of its own, and a massive one, similar in scope to the Astral and Ethereal. This was described as an actual change that occurred historically within Planescape, not as a retcon; something about the confluence of the inner planes triggering the demiplane’s growth, if I remember correctly.
Edition Change: Die, Vecna, Die!
Both 2e and 3e used the Great Wheel cosmology, and so there wasn’t a need, per se, for explaining a cosmology shift (as can be found for the 3e→4e and 4e→5e transitions). However, the ruleset did change a fair amount, and to justify that, we have the Die, Vecna, Die! adventure—an AD&D 2e adventure published by Wizards of the Coast.
Vecna was a lich and a cheating cheater who cheats, and one of his cheats was to escape Ravenloft by becoming a god in Sigil. Those two statements should both be impossible—you can’t leave Ravenloft, and gods aren’t allowed in Sigil. But in Die, Vecna, Die!, Vecna manages to absord the divinity of a demigod coming to Ravenloft, and thereby get Ravenloft to kick him out mid-aptheosis—and got Ravenloft to kick him out into Sigil. The Lady of Pain seeks out adventurers to deal with Vecna, as bringing her power against Vecna after his rule-breaking has so seriously damaged reality could bring down the entire Great Wheel. Said adventurers manage to kick Vecna out of Sigil, and thereby save the day—but a lot of the damage is already done and that’s why the rules are different between 2e and 3e.
World Tree—3e Forgotten Realms only, inconsistently
The World Tree was used... inconsistently, by the Forgotten Realms in 3e, which was weird. It was introduced in the Forgotten Realms Campaign Setting, and originally it appeared to be a massive retcon of FR’s cosmology (which had previously been a part of the Great Wheel like everything else). But then FR supplements sometimes seemed to forget it existed and referenced things in the Great Wheel. Even when things were firm about using the World Tree, they largely begged off from actually developing the cosmology—almost everything in the World Tree just defined each of the planes as being the same as the corresponding plane in the Great Wheel. So in the end there were very limited differences between the Great Wheel and the World Tree anyway, despite Forgotten Realms Campaign Setting having made a big show of retconning everything.
The only really significant difference between the Great Wheel and the World Tree is the absence of inter-planar connections in the World Tree cosmology (e.g. connections from the Abyss to Pandemonium, etc.). The World Tree also defines a few extra planes that easily folded into the Great Wheel planes. Later editions, 4e and 5e, would chalk up the entire Great Wheel/World Tree thing as just being competing mortal theories of the cosmology, not as there having been any real change in the planes.
The primary purpose of the World Tree, as far as I can tell, is to separate the Forgotten Realms from Planescape—that is, to let FR authors be the masters of their own destiny and able to shape their canon as they saw fit, rather than being limited by and having to accept the canon of Planescape. Basically, FR authors seemed to object to FR being “just another world” within Planescape. We’ll see this kind of thing play out through the subsequent changes to the cosmology.
You are also correct that the World Axis was unique to 4e. It was an attempt to simplify D&D’s cosmology (note the large number of planes that were conflated together), and at least in my experience, was one of the least popular things about 4e.
In-character, unlike the World Tree model, the World Axis model is actually recognized as having been a thing for a while, that the planes changed and then changed back. For what it’s worth, this is mostly blamed on affairs going on in the Forgotten Realms, and like most things in FR, it’s mostly Shar’s fault. This would also be the 4e analogue to Die, Vecna, Die! As someone who cares little for FR, and less for the World Axis model, I’m not wild about giving FR so much influence over the wider multiverse, and much happier just eliding the World Axis fiasco out of my campaigns’ history. Don’t need to explain it if it never happened.
But here again we see how there is this tendency, starting in 3e and really picking up steam in 4e and 5e, for the Forgotten Realms to “drive” the rest of D&D’s cosmology. From a Planescape perspective, that doesn’t make any sense—Abeir–Toril is just one, or maybe two, worlds, among many, and many of the big important deities in the Realms are found only there and as such are minor players in Planescape politics. Simply put, Shar shouldn’t have even remotely the power necessary to rearrange the planes like that—to even have a risk of that happening required the combination of the Lady of Pain’s unstoppable force and Vecna’s blatant cheating.
The Feywild and Shadowfell
While most of the World Axis cosmology seems to be unpopular, the Feywild and Shadowfell have been exceptions to that in my experience. The Feywild developed out of the Plane of Faerie, which existed in limited form in 2e and 3e, and the Shadowfell was a conflation of the Plane of Shadow (obviously) as well as the Negative Energy Plane, and also somehow had something to do with dead souls’ journey to the Fugue Plane.
Though 5e has restored the Great Wheel cosmology, the Feywild and Shadowfell still exist.
The Feywild is actually a little awkward for Planescape enthusiasts, as it violates the “Rule of Three” that is so prevalent in the Great Wheel and included the Astral–Ethereal–Shadow triad as one of its most important examples. The addition of the Feywild means it is no longer an example of the Rule of Three. But, nonetheless, the Feywild perseveres mostly because (as far as I can tell) the Feywild itself is pretty cool and popular.
The Shadowfell, on the other hand, has become more of the Plane of Shadow that it once was, no longer a weird stop on dead souls’ journey and the Negative Energy Plane is once again separate. The Plane of Shadow nonetheless remains somewhat negative, in contrast to the Demiplane of Shadow’s original description as being perfectly balanced between positive and negative, and so the term Shadowfell is still used interchangeably with Plane of Shadow.
From the perspective of a Game Designer, there is no difference between an official version of the game and homebrew rules. Pathfinder is nothing more than a homebrew itself from the point-of-view of the people who created it (it is a mod of 3.5, the way that the people who crafted it thought would be better). Do not hold the creators of published worked on too high of a pedestal - they are just people making decisions about what they personally like, just as you are.
You can do whatever you want with the rules of any roleplaying game you play. You can use the rules exactly as written, play without any rules at all, or anything in between, from using a few homebrew rules to a lot of them. What you choose comes down to a matter of preference, as well as questions of Game Design. People made the original rules for a reason, and so you should have a reason to ignore them or to make your own. This can quickly get complicated, and if you are not experienced in Game Design, you may find that you make things far less fun accidentally. Even so, experimentation is the only way to find out what works and what doesn't.