I think the name of the item has morphed into the generic Hamburger or Cheeseburger as a result of the chain explosion. I do recall in the 1960's when I worked for HoJo's in New England the "ER" was not present on the menu. it was HAMBURG, CHEESEBURG and even FRANKFURT. Is that a regionalization?
The only real regional variation of the meat itself that I know of is the "Maid-Rite" concept of Loose Meat on a Bun found in some parts of the Mid-West. The rest of the country uses a Patty...be it round or square, thick or thin, and no matter if it is grilled, fried, steamed or even Deep-Fried.
The place I do see some regional differences are in the condiments, side dressings and presentation of the sandwich.
Is the Bun toasted? Condiments of various flavors on the Bun (top or bottom?) or on the meat? or on the side? What side dressings? Huge list of possibilities here for almost every taste...Fuddruckers style
??. And Lastly, Is it served open to be dressed by the customer, or closed and wrapped in a diaper?
If I am in the mood for a 'local flavor' Burger when we are traveling, then I go for it. If I crave a certain combination of flavors and they are available, then I go for my personal satisfaction. While I enjoy "roadfood" in many forms, I don't consider myself a "Fundamentalist" (apologies to WJ.