Tbf, the distinction (if one is to be made at all) is in the prefix, not the suffix. “Burger” is an acceptable shorthand for both hamburger and cheeseburger, so it could be either. If one is to differentiate, it’s necessary to say the prefix (or to specify toppings to be added or excluded based on what the menu says is standard for this location, i.e “no cheese, no onion, add fried egg.”)
Tbf, the distinction (if one is to be made at all) is in the prefix, not the suffix. “Burger” is an acceptable shorthand for both hamburger and cheeseburger, so it could be either. If one is to differentiate, it’s necessary to say the prefix (or to specify toppings to be added or excluded based on what the menu says is standard for this location, i.e “no cheese, no onion, add fried egg.”)