While we're at it, all the entries for mince (n.) and minced anything in the context of Hack-/gehackt* anything should be marked [Brit.].
The AE term for meat used to make hamburgers or spaghetti sauce is ground (adj.), as in ground meat, ground beef, ground pork, ground turkey ... Hamburger is also a noun that's synonymous with ground meat or ground beef (as well as being the word for the finished sandwich).
We do also have something called chopped beef, which is what Sloppy Joes are made of, the bane of every child's school days. But strangely enough, it's not really chunky in texture, more stringy. Or maybe that was just our school cafeteria.
We don't eat just seasoned ground meat as a dish on its own, as far as I know, so I wouldn't know anything to call it except, well, seasoned ground meat. We would not eat it raw. If we did, it might be called something tartare, but since I have never eaten steak tartare, I don't know if it's served ground or sliced or what.
To the best of my knowledge, even mincemeat in the sense of real meat is known in AE primarily only historically or from books, so that sense should probably be marked BE as well. For us mincemeat is only a tangy-sweet pie filling of raisins, spices and so on.
A recipe might call that a chopped mixture, and for mushrooms the word would also be chopped.