No, that has not been established. What has been definitively established is that role-playing is making decisions as your character would make them, from their perspective; and meta-gaming is (colloquially speaking) making decisions based on information that your character doesn't have, such as specific player knowledge, or the fact that this is a game. If alternative definitions exist, then detractors are extremely reluctant to bring those forward.
But this is off topic. Meta-gaming did not originate with 4E, nor is it an aspect of 4E which was carried into 5E. If you want to drag out that debate, I suggest making a new thread for it.
Sorry, I had meant to say that it had been well established by
everybody else.
The definition of meta-gaming you provide, by the way, curiously does not include aspects of world-building and scene-setting on behalf of the DM, which is where this whole side avenue started to split off in the first place (not deciding on exactly where the pot and the chicken are relative to anything else in the room, for example, is not an example of "making decisions based on information that your character doesn't have").
We also come to the second piece "the fact that this is a game", which brings us to the other part of your thesis, which is "meta-gaming is always bad for the game/community/etc". For one thing, the concept of "Schrodinger's Dungeon" (i.e; nothing exists in the game world until the PCs interact with it, therefore there's nothing wrong with making behind-the-scenes changes in reaction to player actions) is a long-standing piece of DMing advice from forever ago for a reason. It makes perfect sense that this concept doesn't jive well with certain types of players, depending on which
Aesthetics of Play they tend to be attracted to the most (Discovery-seeking players want to know there are things hidden for them to find, and more importantly that they could have missed; Challenge-seekers might scoff at changes made to lighten or strengthen encounters mid-stream; it may or may not impact the immersion that Fantasy-seekers are looking for) but for other players it either won't be a problem or might even enhance their play (looking at you, Expression-seekers; probably also most Narrative-seekers too; and even some Challenge-seekers might actually appreciate some improvisation that turns a cake-walk encounter into something that actually challenges them).
The thing about "acknowledge that this is a game" is that it places the focus and emphasis on the evening on
fun. Which is why there are many conversations on "is meta-gaming always a bad thing?" or "can meta-gaming be a good thing?" I mean, even that other constant nugget of DM-advice, found in at least the 5e DMG, "Try to err on the side of 'yes'? is definitely meta-gaming by your definition and I'm sure many others besides.
Now what that "fun" actually looks like will be different to different players, and certain things that would be loved at some tables would be a complete non-starter at others. And that's a great thing. But the game and the community? Those are only
strengthened by accepting and embracing a diversity of playstyles; rather than gatekeeping based on anybody's own personal OneTrueWay.
I don't think anybody's trying to crap on the way you prefer to play the game. I think everybody's trying to get you to stop crapping on the way they prefer to play.