You have an even more effed-up definition of a hamburger than I could have ever imagined.
Hold on. Let me get my meatless grilled-cheese "burger" off of the skillet. Later today, I'm going to eat my hot ham and Swiss "burger" while my mother eats her chicken "burger."
Speaking of "chicken burger," Chik-fil-a has a long-running advertising campaign (at least 16 years) where cows plead for you to "eat mor chikin" and often discourage you from eating "burgers." Clearly, chicken sandwich != burger. The reason you have "veggie-burgers" is because it is specifically a substitute for beef that intends to replicate beef in what would otherwise be a true "burger." There are many similar tongue-in-cheek references to other meats being a "burger," like a "turkey burger," but there's also "turkey sausage," and so on. By your logic, I can now call beef "pork" because pork = sausage = turkey = burger meat = beef.
Now, an Ichinisan pointed out, there are many burgers that come with sliced bread from a loaf. Being on a bun vs. a slice does not distinguish it from being a burger. An all beef hot dog on a bun probably isn't one because, like a hot Philly Cheese Steak, it isn't *ground* beef and, well, it's only one piece of bread. Ground beef seems to be a requirement for a true "burger, but even a ground beef meatball sub would never be called a "burger" here in the home of the burger.