Michigan, USA here.
hamburger = burger = hamburger meat = burger meat = ground beef. They could all be used in the following sentence: Did you buy any ____ when you went grocery shopping?
The countable form: hamburgers = burgers = ground beef formed into a patty.
If it's another type of meat it specified. Ie turkey burger is ground turkey formed into a patty.
The animal deer is referred to as deer, except when you are talking about the meat then it is referred to as venison, similar to how the animal is a cow but the meat is called beef. You would never say ground cow, or that you ground a cow up, or that you have cow meat, you would always refer to it as beef. It's kind of odd to hear people talking about deer but not using the word venison.
Minced = chopped up very fine, but not ground. We tend to not have minced meat here, and the products they do sell called "mincemeat" I don't think contain any meat, just dried fruit. They are available in a jar in the baking aisle.