We always called it a grilled cheese sandwich whether we fried it in butter or broiled it in the oven. And I've always seen it billed in the menu as a grilled cheese sandwich in all the parts I've lived or travelled frequently...Boston to Florida, New York City, Philly, east to Pittsburgh, and through the South to San Antonio.
We do have a favorite sandwich in our immediate locale here on the tip of the South Jersey shore called a toasted hoagie. That's a hoagie (submarine, grinder) put into a pizza oven until the cheese melts and the roll is slightly crunchy...delicious! Especially on those old "munchie" nights, if you know what I mean. Anybody else have these?