I think the prop gun in that old unknown cowboy movie my family saw back in 1969 might have just been a double-barrel shotgun of some unknown make and model. Why my father referred to it as a "buffalo gun" I will never know but I don't loose sleep over it.
As for BUFFALO GUNS in modern times, I think they are whatever is lawful and ethical to hunt American buffalo. Certainly any gun fit to kill an African buffalo is fit to kill an American buffalo as well. There are expensive two-barrel rifles made in Europe for killing Cape buff and other dangerous game. I think of those as the classic elephant gun.