There's also the scale of the argument to consider. Ok we have "anime is the Japanese stuff" vs "anime is a style". There's also Occam's Razor. The chosen definition should be the one that makes sense, and can be agreed on by the most number of people.
The "country of origin" definition works for 99.9% of cases. There are the occassional questionable ones, e.g. joint productions, but generally if it was produced in Japan with a Japanese dub on release, we can use that as a working, objective definition of anime.
However, style is entirely subjective and ever-changing. How would that work as a "definition"?
- we'd need to list "features of animes", and get people to agree on that list. But can even you come up with a list of things all animes have? Everything on the list would be a "maybe, could have" item.
- we'd need to go over every show in existence, argue about whether it ticks the things on the list.
- then we'd have arguments about some shows which tick some of things but not others.
- We'd have to show how these so-called "universal" features of anime apply to Japanese anime, after doing it for the American stuff, to prove that our definition works. A universal definition isn't allowed to have ifs and buts.
As you can see, rather than "solving" the problem of how Avatar is classified, it would be the biggest clusterfuck in anime fandom history.
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EDIT: The basic problem with the "Avatar is an anime, come fight me" thing is that "anime" means something closer to "sculpture" vs "painting". Anything sculpted is a sculpture, you don't get to be snobby about it because you don't like what they sculpted. If Avatar is anime then so is MLP. If not, then who are the "arbiters of taste" that get the right to declare things as anime.
That video was an example of the Avatar fans I was referring to, when I said Avatar fans want their fave show classed as an anime, but only if they can exclude My Little Pony and other "cartoons".
They're hipsters who want to appropriate the label because it's more "grown up" sounding than "cartoon". But only if they can exclude your "mere cartoons" from the "anime" label. Otherwise they'd be arguing "all cartoons are anime".