Yes. Definitely. Anime as a medium and as an industry has, over the years, influenced many people, films, television series(live-action or animated). Some of these include The Matrix, etc.
The definition of an anime has varied over the years. I consider it to be any animation produced in Japan. So, in that sense, Avatar: The Last Airbender is not an anime. But, there's no denying that there's a lot of influence from anime in that show. If you show a random episode from the series to an otaku(assuming that he hasn't heard of it), he'd probably think it's the dubbed version of an anime.
There are a few arguments people tend to use to differentiate it from cartoons and movies/television series. One being that, anime doesn't necessarily have a lot of moderation, meaning, you can have an anime with blood, gore and sexual content, and the anime gets the appropriate rating for that. You don't see this happening with cartoons. They are all mostly kid-friendly, and are for the most part, non-violent, and certainly don't have sexual content. The only exceptions I can think of are Southpark, and Rick and Morty. The other being that, even though movies don't necessarily have a lot of moderation and restriction(in that if you have adult content, all you have to do is get an R rating), there's the issue of money. CG is extremely expensive, and they can't do whatever they want. Since anime is mostly 2D, and whether or not it's digital or hand-drawn, it's not as expensive as producing a movies.
About the argument of whether anime is very Japanese or not, we can take the example of Hollywood studios trying to adapt anime. Almost every single one of them failed. Why? They didn't accept the fact that the anime sort of feels like something that's Japanese lived-in. Take Dragonball: Evolution or Ghost in the Shell, or even the most recent one, Death Note(which arguably has the least amount of Japanese influence). All three of them tried to give it a western take, and all of them failed, miserably. So, yes, there are some things that you can't just exclude. That's the reason why, I don't think Steins;Gate would translate well, if it's adapted into a Hollywood movie. It's filled with characters who are inbred in the otaku culture in Japan.
The definition of whether it's an art or not is subjective. But, there's no doubt that anime has influenced a lot of work all around the world.