It is an UNDER HAMMER GUN not a SCREW BARREL.[/quote
It could be both, I can't make out a join, it could be rusted over perhaps just in front of the octagon section, but the muzzle shows the deep ratchet like appearance of some screw barrel pistols. The barrel key fits into the deep cuts at the muzzle. This is sometimes mistaken for rifling, and since the pistol has no rear sight its probably not rifled.
I think you are right that the under hammer is an American invention. I think Hopkins and Allen are credited with it.
Its used on many Buggy rifles and larger long range rifles.
I've fired a Hopkins and Allen reproduction .58 caliber once.
It took 300 grains of Black Powder and a one thousand grain two piece slug. hard alloy nose section was swaged to a softer lead base and then wrapped in paper.
It loaded with a false muzzle and a twin lever that clamped on the muzzle and inched the ramrod down slightly with each stroke.
I would guess that the cap is meant to stay on by a tight fit rather than be set in the hammer recess.