Well, personally, I think you're getting confused between the Japanese and English usage of the word. Certainly, in Japanese, the word "anime" simply means "animation", and can be used to refer to any animated series, Japanese in origin or otherwise. But it's been reborrowed into English with a different meaning, and if you're defining it in the context of English, then that's the definition you ought to be using, really. The question of whether "anime" can only refer to series that actually originated in Japan, or whether it can be extended to non-Japanese series that imitate the style of Japanese series, is a more complicated one, but I think it's at the very least clear that in English, "anime" is not a simple synonym of "animation" or "cartoon". That's just not how it's used in English, and never has been.
If you were to ask me whether I think the definition of "anime" should be extended to non-Japanese anime-style series, then personally, I'd say it probably shouldn't. Why? Because in my experience, that's not how the word is used - series of this type are referred to as "anime-style", "Western anime", or similar, but never simply as "anime". And ultimately, the definition of a word is dictated by how it's actually used.
(Oh, and incidentally, you said you agreed on the definition of "anime", but not of "manga" - but the exact same situation exists for both words. "Manga" in Japanese can be used to refer to Western comics as well as the Japanese variety. So if you say "anime" should refer to both, then really, "manga" should as well.)
Guess they need to recalibrate their stereotyping machine.
