I just stick to the country-of-origin side, because the "anime is a style" side has too many caveats - both positive and negative.
If anime is a style separate to cartoons, then it would be logically possible for their to be Japanese cartoons which are not anime. Yet all Japanese cartoons are anime by definition regardless of style. So we're back to "Anime is a style, unless it's Japanese, in which case style is irrelevant to whether it's anime".
If we're going to dictate what stylistic aspects "real" anime have, then we need to explain why we are selecting those particular shows as "real" anime, and what is "not real anime" about the Japanese shows we've excluded. So are we talking similarity to 60's anime, 80's anime, modern anime or what? Disney is anime if you go by Osamu Tezuka. Powerpuff Girls is anime if you go by Panty & Stocking. Hanna Babera shows are anime if you go by something like Doraemon. So which exact of the separate anime traditions do we count as "real"? Which is all sort of cultural imperialism right there. I recall in Australia some white art experts telling Aboriginal artist that their works not "real" Aboriginal Art, because they've incorporated other ideas into it.
The problem is most likely we lack the vocabulary to properly classify styles, so we lump disparate styles into the term "anime", even though they don't really all fit in that label. What would be more reasonable would be to appropriate some of the demographic/genre terms and apply them to western works. e.g. saying "Avatar TLA is the American take on a Shonen anime" is much more accurate. Like calling original works made by the producers of Power Rangers as Tokusatsu shows, when they follow all the conventions, is pretty accurate.