Well, I'm just guesing here because I have no documentation for what I am about to suggest. My feeling is that the term fowler was used to distinquish a sporting arm from a musket, when both arms were smoothbored, and roundball rifles were relatively rare and "regionalized". In short I think the term fowler was simply common parlance for a smoothbore sporting arm.
By the time of the American Civil War, not only was the typical musket a rifled weapon, but rifles were common everywhere, and the term shotgun was employed to distinquish a sporting arm that wasn't rifled.
But it's just a guess, and I'd be curious if someone can document the transition in terms.