@Omicron: as others have said, going back to Nibelheim now that Cloud has his true memories back to explore it as thoroughly as you did when you collected Vincent will provide a new block of completely optional, highly important lore. I think some people might argue this is more crucial than the Icicle Inn, although I personally don't think so. Still, of the optional lore in the game, it's certainly in the top two spots. (There's more than two optional lore-dumps in the game; Gongaga was also one, and there's at least one more you'll need to do a lot of exploring to actually find).
The subsequent adaptations don't agree with each other at all where the deed happened, what order things happened, or even whether Sephiroth was thrown or jumped.
This almost feels like unconnected teams writing the same thing at the same time, but wasn't ff7 made before the rest were even started, by a long shot?
Now, I will preface this by saying this is my personal theory, and so nobody is forced to agree with it. It's just opinion and nothing more.
That said, I think that the writers of the compilation just fell for the very same "invincible Sephiroth" propaganda that Cloud fell for in-game, for Sephiroth's "mystique". Thus, they felt that him being taken down by Cloud was not properly fitting such a great individual, and tried to rewrite it so he came across less humiliatingly stupid in the scene. Whereas in the game itself, this is the point where Sephiroth's image of being this invincible badass is shattered, and he's revealed to be arrogant and, in hindsight, incredibly petty.
The whole point of this sequence having Cloud kill Sephiroth, instead of something else happening (such as Zack doing the job, for example) is that Sephiroth thinks himself the chosen one, above the common man. As a direct result of this arrogance, he left himself open to be taken down by somebody who he'd not merely dismissed, but also made as angry and determined to hurt him as somebody can be, by killing his hometown, his only family, as well as wounding (possibly lethally, for all Cloud knew!) the girl he loved and his best friend. Sephiroth didn't have to do any of that; he could have made his way to Jenova in secret, used his connection to the clones to build himself an army over time, and if he had to kill the mortals, do it in a way that left no survivors. He didn't do that, because he arrogantly thought they didn't matter. He was wrong.
And then, after he was stabbed, and could have slunk away to lick his wounds and recover, instead he came out to confront the one who had insulted him so, secure in his arrogance that he'd only been hurt because it was a backstab that took him by surprise. Instead, his strength fails him because he wasted time gloating (that's how I chose to read the "Cloud leveraged himself to the ground", as Cloud's grip on the Masamune and wiggling on top of it changing the balance point enough that Sephiroth's wounded strength wasn't enough to keep him suspended), and gave the person who had all that anger motivating him the time for the one burst of strength needed for him to be thrown in the Lifestream.
Plus, in addition to completely removing all the coolness Sephiroth has been cloaked with the whole story, this also makes the point that it didn't take a super-soldier enhanced with Mako or Jenova or whatever to defeat Sephiroth; it just took somebody who had been wronged enough to not hesitate to strike when the opportunity for payback presented itself, and who had been wronged enough to push himself to perform past human limits just one time. This is important, too, because it makes it clear that it's not Hojo's experiments that made Cloud Sephiroth's match, it was just that he cared enough about the people Sephiroth was hurting to do the impossible for them. That matters to the themes of the story a whole lot.
Obviously, I deeply dislike that the writers have been trying to retcon this away in the following adaptations of the story, both because it completely misses the point of the scene, and because it takes away what is one of Cloud's most important and unambiguous moments of awesome. It's Sephiroth's hype managing to overwhelm the story, because writers got caught up in how cool he was that they weren't able to stomach leaving intact the scene where it's proven, beyond any shadow of a doubt, that he's really not. So they rage about it by rewriting it.
Just like Sephiroth himself does when he speaks to Cloud. I think there's a lesson to take in that, but this is pretentious enough as it is, so I'll stop here. I'll just reiterate that I think the writer are rewriting the story on purpose, because they either don't get it, or don't
want to get it. Everybody can make their own judgment over that.