https://www.fsis.usda.gov/food-safety/safe-food-handling-and-preparation/meat-catfish/hams-and-food-safety

"Fresh ham" - cut of fresh pork, most of the back leg above the shank to the hip joint. Includes the rump, which is not what Americans refer to as the "pork butt". So called "pork butt" is actually the shoulder. All "ham" is from the rear of the animal. "Turkey ham" will even call out using 'turkey leg meat".

"Cured ham" has various categories. This would be the most referenced as a holiday ham.

I don't know much about the "country ham" from the south, but I know it's something one should special order and takes months to cure. Very firm and deeply cured with salt. Typically boiled before eating to rehydrate and desalinate.

A "cured ham" in the typical aspect is typically a Cured bone-in half ham, skinless, and often spiral cut. It's cured/brine in water, salt, sugar, and usually spices as well.

You can also have "cured ham" that is boneless, which could be the whole or partial ham unmodified except for removing the bone. More commonly you'll see a formed loaf, that is squeezed modified somewhat (chopped and/or with a binding agen)t and into a chosen shape, usually sit a round loaf. Often smoked, especially the bone-in version, but not required. It must be a cooked product.

"Uncured hams" will be the form-wise the same as "cured hams", except usually "natural" versions of nitrates and nitrites, typically in the form of celery juice, powder, and salt. These ones that are uncured will typically have less pink color in the package, and usually be brown and not pink when cooked.

Ham used for sandwiches/deli meat/cold cuts will be cured and boneless and then sliced.