Cartoons, while a broad media term, do not include anime as it is its own category
There's a separation between cartoons and anime that, while similar, keep them in their individual categories. As an example, Pokemon is an extremely popular anime that's been going on for 24 years. Created in Japan and marketed mainly as a game with an anime. However, in America it's seen and almost entirely marketed as a game and cartoon. Part of the reason for this is the age range. When animes are dubbed for American television the age range is usually the defining factor of whether it'll be promoted as a cartoon on or an anime. So while shows like Pokemon and Bakugan are age appropriate, shows like Naruto and Bleach are usually shows that end up on Adult Swim or have a higher maturity rating in America. Fighting, blood, and crude language is usually the defining factor in anime that keeps the show on Adult Swim. Whether the shows are meant for a younger or older audience does not change the fact that they are still anime though. While cartoons might sometime refer to anime, anime never refers to cartoons.
In this debate between if anime and cartoons are the same thing or not, what also has to be questioned is the angle this question takes. They are both animations that follow the same process of story, story boarding, animation, then to the screen. Comparing their similarities and differences they are more similar than they are different. What has to be asked is 'does animation need to be referred to in subcategories?' While it sounds a bit confusing, animation is a term that encompasses many art styles and art forms. Many things can fall under 'animation'. Flip books are a type of animation though it might take a moment to make that connection. These differences in the creation stages separate what a persons creating. The story, the image, the meaning someone's trying to convey is all part of that process from getting an idea onto the screen. Categories and subcategories keep things straight in the developmental and marketing stages but in the end they're animations.
[P1] America usually separates anime based on maturity which is why some animes are called cartoons, but they are still anime