So, I eat two different things, both of which I call a “grilled cheese sandwich”.
One is what I can buy at a diner. It has two pieces of bread, with cheese (or sometimes, cheese and something else, like ham or tomato) in between. The bread is smeared in something greasy, like butter or sometimes mayonnaise, and cooked on a grill (griddle). Both sides are grilled, and the whole thing is also pressed together with a spatula as it cooks. In this one, all the heat comes from the bottom, and the sandwich is flipped so both sides get grilled.
The other is what I make at home in my toaster oven, and is what my mother used to make for me. It’s a piece of lightly toasted bread, with slices of cheese on top (sometimes with tomato) that’s put back into the toaster oven until the cheese melts. I watch it carefully and pull it from the oven before the molten cheese drips off the toast and makes a mess. No butter (nor any other non-cheese source of fat) is used. It’s an open-faced sandwich, and is not squished. The toaster oven actually has heating elements on both the bottom and the top.