OK, so I started something at FB, but it needs more explanation. This may clear up a lot of transatlantic confusion.
In Britain the word grill is equivalent to the North American word broil. We can argue until we are blue in the face that this is wrong, that a grill actually means the metal apparatus used to separate food from its heat source (think of the bars on a window, also technically a grill) ultimately derived from the word griddle, from a latin term meaning "latticework". But how it is used NOW is what matters.
What grill means NOW is dry heat, i.e. to toast something. It can be a barbecue grill, or a waffle pan (although these are often actually greased, see below) and all sorts of things in-between. It can be above or below (or both) or at the side. It can be a long fork over an open fire, really.
Even then there is a looseness in these definitions, as North Americans fry a sandwich and call it grilled, and restaurants have a large, solid, flat hot surface which can be called a grill, griddle, and other things.
There is no wrong or right in this. Word usage changes over time and from place to place, that's just how it is. The local term is correct, locally.
But because we discuss food, coming as we do from all corners of the globe, it can get very confusing.
Frying, generally, means to use fat (oil etc) to cook food. It can be just a little fat (a fried egg) or a whole lot (deep fat, turkey fryer). However, the french term sauté is often used to describe gentle frying. I think this has bercome a popular term simply because people have equated frying with bad (unhealthy or unpleasantly greasy) food. But it's more of a euphemism than a difference.
So, when a piece of bread is cooked with a little fat in a hot, solid, flat pan, strictly speaking it is being fried, but it tends to be called a grilled sandwich. People are funny that way. In Britain this might be called a toasted sandwich, and in France "un toastie". The French think this sounds cool. In revenge we have started making croque monsieurs. It's all jolly good fun.
There are pans and small appliances on the market on both sides of the Atlantic that are used in this way, over heat on a stove, or with its own independant heat source, that are called grill pans, or grillers, etc etc. Adds to the confusion.
Therefore the best we can do is define the end DISH.
A grilled sandwich, a toasted sandwich, un toastie, or a croque monsieur involves TWO slices of bread with a filling, and the bread is FRIED. Call it what you like, that's what happens. It might be fried in a frying pan, or in a special electric press, possibly even a panini press. I'm not even going to start on the panini debate. We'll just leave it at that.
But cheese on toast is ONE slice of bread, and it is toasted, not fried. The cheese is melted right onto it using the same dry heat. Doesn't matter how.