Anime
While it is clear that the word anime derives from a western term, there is some debate as to its exact origin. One very common interpretation is that it comes from the French term for cartoon, "dessin anim", which translates literally to "animated drawing". However, others believe it's simply a contraction of the word "animation" as pronounced in English.
While the French origin is perhaps the most widely spread theory, it may simply be a myth related to the popularity of anime in France in the late 70s and the 80s which was so spectacular it even led to some French-Japanese co-productions. But animation in Japan has existed for longer than that, and the full word for animation there is (animeshon) which follows the English pronunciation, not the French one. Combined with the fact that Japanese have a habit of incorporating English words into their language much more often than other western languages, this makes a strong case that "anime" is actually the short form of the Japanization of the English word "animation."
There is much controversy as to whether or not the label "anime" can be extended to non-Japanese animation that looks like Japanese animation. Whether or not the label can be extended to non-Japanese animation depends largely on the technical definition of "anime" that one choses to accept:
Defined by origin: Defining "anime" as animation produced in Japan allows for a fairly black and white application of the label. The only gray area occurs with co-productions that may have had a portion of their animation, and/or scripting produced outside of Japan.
Defined by style: Defining "anime" as a style of animation that originated in Japan is much more complicated, however this definition would allow animation produced outside of Japan, but conforming to the defined "style" to be called "anime." It is difficult to determine if this "style" should be determined solely on drawing style (ie: Big eyes, small mouth, pointy hair), if it should include editing techniques (Japanese animation typically makes more use of "cuts" and "camera angles" than most non Japanese animation), and whether the narrative or storytelling style should be included in the definition. Perhaps the biggest pitfall of this definition is that, due to the wide variety of Japanese animation, regardless of any style based definition, there will always be Japanese animation that would not fit the definition, creating a scenario where some Japanese animation would not be anime.