In advertising jargon of the 1830 era, a "Fowling Piece", or "Fowler" was a single barreled smoothbore gun made for firing shot, and a "Shotgun" was a double barreled gun for shooting shot.
The line is not clear-cut, but as a general rule, that's how they were referred to at that time.
I cannot say what terms were used in the 1770s, but I have never seen the word "Shotgun" used in any letters, adds, or bills much earlier then about 1820.
If I am in error here, I'd welcome additional information myself.