Perhaps it depends upon whether you want little bread with your cheese, or you want a little cheese with your bread. a sanfwich may have to have some type of mayo, or not.

When my niece was young (pre-teen, I think) and the family was in Rome, she ordered a grilled cheese. That’s what she got ... cheese that had been grilled, frico. Not at all what she or her parents thought she was going to get. We still talk about ”Roman grilled cheese.”

I love grilled cheese sandwiches. I especially love them at restaurants because they are so much better than what I make at home. At home, mine are very pedestrian: white bread, Kraft Deli Deluxe single, and butter. That’s it. It is the answer to being hungry but not wanting to put any effort into preparing something.

My all time favorite was a local place that served a grilled cheese with brisket on it. The bread was sturdy and thick sliced, the cheese was shredded and had a good bite, and the brisket was melt-in-your-mouth tender. When the price of beef went up, they took it off the menu. Their IG post said they would have to charge $30/sandwich to break even and they did not believe anyone would pay that. Right now I want one so badly, I might pay that.

"Boulong's Cheese" has all your answers.

Dave, I get your point about the mayo, but a lot of people use it on the outside of a grilled cheese because it crisps up so nicely. I don't put mayonnaise on a cheese sandwich, though I might on a meat and cheese on the opposite side from the cheese, and usually where the vegetables are. My regular size quesadillas are dry grilled (or steamed if I'm being lazy), and if they have sauce, it's spicy. ;)

American quesadillas are quite different from Mexican quesadillas, to start with. In the U.S., a "quesadilla" would be made with a flour tortilla (rare in Mexico, except northern Mexico) that is grilled on cast iron or else in just a plain skillet.

In central and southern Mexico, a quesadilla is made with fresh masa (no wheat flour) that is formed into a disc, filled with cheese and often vegetables, especially huitlacoche, sealed, and then deep fried in what looks like a wok. Here I am waiting for a huitlacoche quesadilla in Mexico City:

I've never had flour tortillas in Mexico, but the northernmost place I've stayed in Mexico is Cabo San Lucas. They probably have flour tortillas there, since it is extremely Americanized, to the point where you do not need to speak Spanish there, but the emphasis on food is seafood. They probably also make American style quesadillas, to cater to the tourists, most of whom do not speak Spanish.

I make both types of quesadillas at home, but of course the ones with flour tortillas are much easier, since they are grilled instead of deep fried. I often add ham or turkey breast, thinly sliced mushrooms, and Provolone cheese, as well as black pepper. I do not consider it particularly Mexican (if at all), but I do like it, and it is quick and easy to make. It is common in Mexico to have vegetables (especially mushrooms or fungus) in quesadillas, but sometimes meat is added as well.

When I add pasta sauce to my flour tortilla quesadillas, I no longer call them quesadillas - they are more like mini pizzas.

I never make grilled cheese sandwiches, unless you count Reuben sandwiches, which I make often, using rye bread that I make, my own sauerkraut, and my own mustard. I do not make the cheese and use Swiss, Gruyère, Emmenthaler, or Jarlsberg. I tend to have Jarlsberg most often, and so that is what I generally use, and I like it better than the Swiss cheeses that I normally find.

I am not that hung up on names, and so it really does not matter to me if someone calls a sandwich that has cheese and is grilled a grilled cheese sandwich. I am also flexible with the name "quesadilla," although I make a distinction between Mexican quesadillas (which resemble empanadas) and American quesadillas.

However, I do not call anything made with a flour tortilla a taco. That is just too far from anything I have ever experienced.

In Britain, we have a toasted cheese sarnie which is 2 slices of bread with grated cheese in the middle and toasted in a pan or under the grill.

And we have cheese on toast. Which is grated cheese on top of a slice of toasted bread then grilled until melted.

Then, we have Welsh rarebit which is a cheese sauce made with cheese melted into beer, with a dash of worcester sauce and a spoon of mustard, grilled on toast until bubbly.

The French have croque Monsieur which is a toasted cheese and ham sarnie or a croque Madame which is a toasted cheese and ham sarnie with a fried egg on top.