Defimitions: (I'm from Michigan)

Hoagie - Two or more kinds of lunchmeat and cheese on a hotdog bun, usually heated.

Sub (Submarine Sandwich) - Pretty much anything on a long bread bun or part of a long
bread loaf always with lettuce and other salad acompaniments, never heated.

Grinder - basically a sub on which the bread/bun was buttered with butter and/or
mayonase half of the fillings on each half of the bun and run face up through the
broiler before the salad is added. Obviously always served hot