I definitely remember when the new Sonic Adventure designs were revealed and people were either totally for it, or absolutely against it. At the time, I thought it was pretty neat, but I think the age I was (around 11 or 12) I was the target demographic of who they were aiming at with the new "cool" design philosophies. But I do remember it immediately being a sore point amongst some Sonic fans of the time, so I would argue that as soon they gave Sonic green eyes and longer limbs and spines there was a split between "new Sonic" and "classic Sonic". To further my point that it's mostly born of a design change and not so much gameplay differences, you got Mario. Mario Games have definitely changed over the years since his original outings, in much the same way Sonic games have, but there is no "classic Mario" or "modern Mario". It's all just Mario. And I think it's because, more or less, Mario's design has stayed consistent. I think modern Sonic came about because of the times. Sega didn't think Sonic itself was relevant anymore and wanted to revitalize his presence and felt a total redesign would achieve that. And after the times changed again and the "modern Sonic" they came up with isn't really all that "cool" any more, they're trying to figure out how to reconcile the two designs to please everyone.
I still think they were in the right for the redesign. Crash, Spyro and the like had definitely upped the "attitude" of the later 90s to where classic Sonic seemed cutesy or quaint (more akin to Croc or Banjo); an idea that only amplified as games got more brooding and serious into the 00s. And honestly to this day IDK if that's got any better. The classic design maybe over-cutesified by Sega now, but even at it's base people remember it more endearingly than they do "wow, what a cool character". Modern Sonic isn't exactly "peak cool" anymore either, mind, but it can fit to content as cool or cute as they need, plus it can hold a neutral tone that I think Sega finds flexible (in a similar way that Nintendo sees Mario). Personally, I think classic could be just as cool as modern could (if not way cooler), but it also requires high quality animation and spectacle which they can't always afford. Classic Sonic is almost always coming off cool in spite of his appearance, and I think that's true even in the old days. Look no further for that than Sonic Jam. He's adorable! I digress because this isn't what the topic is about, but it's interesting to think about.
I think that the terms "Classic" and "Modern" would eventually come out spontaneously when there is a well defined formula that can identify a specific part of a series (and, in the case of "Modern", when such formula is the current, active one). The original classic trilogy and CD all have similar gameplay and look. Modern Sonic instead is much more inconsistent in its style, so the definition is often used to mean "not classic", or specifically refered to the details of the character design (green eyes, different shoes) and timeline placement (set in the future compared to the classic games) rather than gameplay or overall formula: fans in fact differentiate Adventure Sonic, Boost Sonic, and several other sub-groups of games depending on the little variations and things in common that those games have. I like Kirby games as well, and I know that occasionally people in the fanbase call the post-wii games "Modern Kirby", because all those games look similar to each other. It's a lot more unusual to see the old Kirby games getting called "Classic Kirby", they are almost never called that way because of how inconsistent the older games were; often they're divided in groups like fans do with Modern Sonic: Sakurai games, Shimomura games/Dark Matter trilogy, Flagship games. We don't see those terms used for Mario, because every Mario game is different in a way or another... the closest thing was the "New Super Mario Bros" sub-series, but still, at the time there were also Galaxy, 3D Land, and so. Zelda fans are only recently starting to identify the older Zelda games as "traditional Zelda", "old formula" or something, opposed to the recent open world games, but it's not yet a commonly accepted term, just some people starting to draw a line between different groups of games. So, in conclusion, I don't know who started the definition of Classic Sonic, but I can see more or less how it originated, and it's a pretty natural thing that happens in every series with a recurring formula and changes at some point in their history. Sorry for bringing this sentence out of context but yeah I agree that the split should not exist, at least not officially.
