In the US, "Burger" generally refers to ground meat patty, "hamburger" can mean either the sandwich or the ground beef itself. If you bought a pound of ground beef at the store, you might say you bought "a pound of hamburger." Turkey burgers, chicken burgers, bison burgers, etc all focus on the idea that it's the ground meat patty that makes them a burger. Most Americans (or at least me) would say a patty melt is a type of burger, because it has beef burger patties, despite not being on a bun.
So chicken sandwiches, pork sandwiches, and steak sandwiches, which are all common here and typically served on buns, aren't considered burgers because the meat isn't ground.
It's so thoroughly thought of this way here that that the product "hamburger helper" doesn't involve bread or sandwiches whatsoever: it's a box of pre-portioned ingredients, maybe with pasta or rice, that you cook with with ground beef in a pot or skillet.
Edit: old timey American cartoons used to refer to "hamburger sandwiches" which were by all accounts the same as burgers today; the "sandwich" part got lost sometime before it got to Australia, i guess.
