A burger implies the meat is made out of beef. If you replace the main meat ingredient of a burger with, let's say... turkey. Then by definition it's a TURKEY burger because it contains no beef.

If you take a hot dog and replace the sausage with a chorizo, is it still called a hot dog?

If you make a meatloaf and instead of meat you use tofu... is it still a meatloaf?

So no, veggie burgers should not be called just burgers.

Should you distinguish between candy and sugar free candy? It's important to have distinction when you are representing something and there is the assumption that a burger is in fact meat. Any alternative to that assumption should have notification.

This is a slippery slope. We start calling them burgers and next thing you know, they'll want to call everything else a burger. By lumping everything together and using one word, will get confusing. I mean, what will I tell my children? Wait..., this argument sounds familiar ;)

Burger in most people's minds implies beef, not veggie. Therefore, it should be noted as such.