What is a grill to you, then? Is it the same as a griddle?
Griddle = frying pan. A “grill” is the same idea but on a much larger surface. A place called a “bar and grill” will have beer on tap and, back in the kitchen, an enormous flat table-like frying pan, like what McDonald’s uses to cook their burgers. Capiche?
Just added the bold bit, as a “grill” can also mean, for example, a Weber kettle.
Heh, “divided by a common language” in a way that actually influences a recipe - to us, a grill is a contraption that radiates heat from above - so when we make “grilled cheese” we put it “under the grill”. I wonder if “grilled cheese” came from the US, and we misinterpreted how to make it? (It’s very tasty the way we make it, too, especially with Worcestershire sauce. And probably a tad less fatty.) How would you do a tuna melt?
Hm, I’d call your method of grilling cheese “broiling cheese”. Sounds like they’re open-faced? And that’s how I make a tuna melt - on a cookie sheet in the oven, then set it to broil.
Here a grilled cheese the “real” way two slices of bread buttered on the outside, cheese in the middle, then cooked in a frying pan until golden. Unless you’re lazy, in which case one can put cheese between toasted bread slices and nuke it until the cheese runs out. 
OO - I love discussing both food AND English-English/Aerican-English language differences! There may even be some regional variations in Americxan-ENglish which will complicate definitions.
Cooked by heat/flame coming from above the food: in American-English = BROILED
Cooked by heat/flame coming from below the food while the food is sitting on a metal grate: in American-English = GRILLED
Unless you’re talking about GRILLED CHEESE, which where I come from is usually done on the stovetop in a dry frying pan. The bread is buttered on the outside so it browns & crisps up. Sometimes also called TOASTED CHEESE even though there is no toasted involved.
Tuna Melt in my part of the world = two slices of bread, buttered on the outside. Tunafish (being tuna mixed with mayonnaise and if you’re fancy some chopped celery and onion) and slices of cheese between the slices of bread (swiss or american are popular choices). Dry-fried like the aforementioned “grilled” cheese in a skillet on the stovetop.
Yeah, grilled cheese is usually open-faced here.
But now I’d doubting myself - maybe we usually say “toasted cheese”. This does involve toast, though, as you have to partially toast the bread before you throw it under the grill (broiler?) otherwise it ends up soggy.
I suppose the real inventors of the grilled cheese/toasted cheese would be the French as usual - with the croque monsieur. Will have to look up how they make that properly, though I think it’s grilled (broiled).
Presuming you have a tuna salad handy (can o’tuna, drained; chopped onions, mustard, mayonnaise, sweet pickle relish, or chopped dill pickle, seasonings to taste, and mixed to the consistency you prefer), you make a tuna salad sandwich, with a slice of cheese (no lettuce or tomato), butter the outsides of the bread, place it in a medium hot skillet, and toast it on both sides until the cheese is melted.
So. tou you guys over there, when you order up a grilled cheese sandwich, it’s broiled. Let’s say the chef makes a cheese sandwich, butters the outside, and browns it in a skillet untio the cheese is melted and the outsides are brown. What do you call it? Toasted cheese?
If you’re really going to make an Elvis-style sandwich, don’t you have to fly to another city first? Or is that just an urban legend?
Cooked in a skillet (which we call a “frying pan”) with butter - to me that’s not “toasted”, that’s “fried”.
A fried cheese sandwich?