"I get what you're saying, but I don't get what you're saying."
Does this help? Cartoon is a style of animation basically exclusive to the west, whereas anime just refers to the broad world of Japanese animation. For example, Elfen Lied could never be considered a cartoon. When you hear the word "cartoon", people being ripped limb from limb in a bloody mess of body parts is not what you would think of. I mean, if you want to keep being wrong, have at it, but the two concepts are different things with cartoons being rather particular.
I guess the closest thing I can think of would be like saying frozen yogurt is ice cream too, despite the fact that they are similar, but not the same thing and would not be referred to as the same thing.