I have seen as well as noticed a number of Western animation series (including American) that are created and produced in the style of anime/anime-style

The dialect?

^ No the origin; where it was created.

2weirdalfan My thoughts exactly.

Yes and no. I believe the term anime in itself implies a reference to a Japanese product, why else take a Japanese word for what we could describe perfectly well with our respective native languages. Though in my opinion this idea has shifted more toward a pursuit of an established concept which originated in Japan instead of the interpretation that an anime must have been created by and for Japanese people.
Therefore the usage of the word anime might simply display Western cultures' acknowledgement of the roots of this kind of entertainment.

I think it must be Japanese to officially be classed as an anime, but it doesn't have to be Japanese to be GOOD. Avatar: The Last Airbender was amazing. I found the sequel (Legend of Korra) very disappointing, though. :(

I now understand there are plenty of other series that are STYLED in/after anime and overall are ANIME-STYLED series

Well, 'anime' itself is a reference to Japanese cartoons, involving a particular style of animation, which is heavily detailed and involves many special features, as well as typical features, like the random simplistic style every now and again in anime where it's usully detailed (FMA always does this. Lol) and the anime sweatdrop thing, when the character is exasperated.

Basically, being technical, anime has to be Japanese, because that is it's origin. However, it is true that a lot of cartoons base their style on Japanese 'anime' and try to simulate the same effects. I can't say that I have seen any cartoons like that, myself, though I'm sure plenty exist.