Oh but you know better than all of us do you?
Edit: I'll give you my explanation of what it means, after some consideration, it is typically translated over to animation, but thats not what it means, it is also not the japanese for cartoons, anime is the word used for japanese cartoons drawn in a japanese style for japanese people. Most japanese cartoons are anime but not neccesarily all since not all artists in japan like the japanese drawing style but prefer the western one. Anime doesn't mean anything particular, its just a word for a specific category of cartoons and animated 3d movies and shows.
I won't call a american cartoon drawn in anime style anime just because of the style, it just wouldn't feel right to me. because if it isn't made by japanese people with japanese audio it doesn't have the right to be called anime.
but if you're not convinced heres something for you to read. http://en.wikipedia.org/wiki/Anime
Great, you just wasted your time by letting the point fly right past your head.
"Anime fighting". Oh like what? Like how they fight in Ghost in the Shell (mostly realistic)? Like how they fight in Inu-Yasha (totally unrealistic and insane with big swords)? Like how they fight in Gundam, with huge lumbering robots (or, on the flip side, in other Gundam series where the Gundams are freaking insanely fast)?
What DOES "anime fighting" mean? The answer is it doesn't mean anything but what someone's narrow, unrealistic view on the medium as a whole is. You do not have "anime style fighting". You might has "shonen style fighting", or perhaps "final fantasy 7" style fighting. You do not have an anime style fight in the same way you can't have a cartoon style fight or a movie style fight or a book style fight. By claiming that there is an "anime style" of fighting, one is seriously mislabeling an entire medium of animation.
tl;dr: That word isn't being used properly and is insulting an entire medium of animation.