The term is used for both nowadays.

The traditional definition of “croque monsieur” is that of cooked sandwich; as the cooking is of a special sort that involves the pressing of the two sides of the sandwich one against the other by means of two small and curved hot plates, the result being, according to the ingrédients, to preserve a steamy and moist interior to the sandwich when cooked, as well as giving it the shape of this mould formed by the plates, it is difficult to have anything else than bread on top; in any case, cheese would stick to the hot plate.

A second sort of croque-monsieur is topped with a fried egg once cooked and that one is happily called a croque-madame.

However, similar sandwiches are made nowadays but with this difference that you can find grilled cheese on the top of them, and they are also, unfortunately, called « croque-monsieur ». You can see pictures of those below.

There also vegan croque-monsieur and getting a little bit carried away by their imagination restauration people have called those croque-mademoiselle.

note : Traditionnellement considéré comme invariable. Les recommandations de l’Académie française de 1990 admettent la forme « croquemonsieur » et les pluriels « croquemonsieurs » et « croque-monsieurs ».
Le très rare pluriel « croque-messieurs » se rencontre aussi et a été décrit.

However, the term “sandwich au fromage grillé ” does exist and it is a translation of the american “grilled cheese sandwich”; the synonym “grilled cheese” which I have no idea how to pronounce, is also used; this can be verified here, where we also learn that these sandwiches can be made by means of an apparatus for cooking croque-monsieur (two hot plates). It is surprising though, considering that there is nothing on top (the cheese is inside): this is really a croque-monsieur, as croque-monsieur are also made with frying pans. Nevertheless, it seems that this is the fundamental definition of grilled cheese sandwich as seen here (free encyclopedia); it is not, as the definition you know tells (possibly a recent addition) what you call an open face sandwich but a true sandwich (cooked), which really is a croque-monsieur.

It does not have a French equivalent because there is no grilled-cheese sandwich in French cuisine. In Quebec they simply say Sandwich au fromage grillé ou Sandwich grillé qu fromage.