The Japanese consider Western cartoons anime (including, but not limited to, Disney films, etc.). Since the Japanese invented the word "anime," you can't really argue with them that they're using their own word incorrectly. If they say that Western cartoons are anime, then cartoon = anime. And since "anime" is just a shortening of the English-invented word "animation," which is one of the definitions of "cartoon," it can be argued that all anime are cartoons and all cartoons are anime.
However...
The Japanese have taken English words and perverted their meanings to mean something that's completely different from their original English meaning. Take, for instance, the word ????? (manshon), which is taken from the English word "mansion." To us, "mansion" means a large home or manor, usually occupied by a wealthy person and his/her family. To the Japanese, "manshon" is an apartment complex. We could argue that they're using the word incorrectly, but they've taken the word "mansion" and made it their own, with their own meaning. The same could be said for the word "anime." Westerners have taken the word "anime" and interpreted it to refer to the particular style of Japanese animation.
Even if the Japanese say "anime" is the same thing as "cartoon," the word has taken on a whole new meaning in the English language. If you study the etymology of any word, you'll find that often their meanings have evolved to mean something different from their original meaning. I think this tends to happen a lot when words from a source language find their way into a new language and become mainstream in the new language. At least, I can say it's true for English words that make their way into the Japanese language.
My take on it is that you can say "anime" and "cartoon" are one and the same, or you could say that "anime" specifically refers to the Japanese style of animation, and either way you'd be right. I don't like to quote dictionaries because languages are constantly changing and evolving. To me it doesn't make a difference. I think those who argue that it can only be one way or the other are the ones who are wrong.