In Europe/America -> Anime = Japanese Animation (or any other east asian country) In Japan -> Anime = Cartoon
That is it. nothing so complex behind it.
Actually that's wrong Cartoon is a word from Italian that means a sketch. Typically use as a prototype drawing for master works. There are cartoon by Michaelangelo for the Sixteenth Chapel as an example. When English adapted the use of the word, they used it for comic they drew in news print. When it came to animation pioneers, like Disney, they refereed to the animation as Cartoon Animation, which was later just shortened to Cartoon. Technically cartoon has nothing to do with animation at all, rather the style of what is being animated.
Now Anime is completely different. It's origin is the word Animation. It evolved from the full Japanese term Manga Animation, which as shortened to Anime. It specifically was meant to refer to manga being animated, but due to the way it was shortened it could be use for all forms of animation. However when English cross-back adopted it, it specifically meant Japanese style animation, which technically is close enough to it's source term manga animation that there is no notable deviation in meaning.
What the point here? Cartoon's are a specific style of animation, Anime are a specific region's style of animation. An anime is not a cartoon, but a cartoon could be an anime. You can't cross use the terms in English like you think. Since anime is based on manga style, which has it's true roots in old sumi-e drawing of ancient Japan and specifically the Tales of Genji scroll. Where as cartoons are base on Renaissance scribble art.
Ah you are probably right, I like to keep things simple mostly :P. Though it does get confusing, What would avatar - the last airbender be? because I remember it getting a reward for Best Cartoon. somewhere though most would consider it an Anime I think.