The first three 44-gun frigates build for the newly-established U.S. Navy were the
Constitution, United States, and
President. All three were, or at least were supposed to be, built to the same hull lines, which were drawn by Joshua Humphreys.
According to the references I've got, the
United States was the one with the raised poop (or, as some contemporary documents call it, "round house"). She was the first one launched, and apparently was intended to serve as a flagship, with a supernumerary commodore on board. For a while Revell was marketing a
United States kit, which was a reissue of the
Constitution with the addition of the raised poop (and, I assume, some changes of detail - such as the name on the transom).
As I understand it there are two reliable contemporary graphic sources for these ships' appearance. The first is Humphreys' original draft of their hull lines. It's rather crude and simplified (he wasn't much of a draftsman), but it's extremely useful to modelers interested in the ships' "as-built" configuration. The second is a set of "Admiralty draughts) of the
President made by the British after they captured her. These drawings are much more detailed, and beautifully executed. They can be relied upon to tell us what that ship, at least, looked like during the latter part of the War of 1812.
The Revell
Constitution kit is based on a set of plans that were drawn for the Smithsonian Institution, back in (I think) the 1950s, by George Campbell. I have the impression that he used three basic sources: the two drawings I just mentioned and the "Isaac Hull model" of the
Constitution in the Peabody Museum (now the Peabody-Essex Museum) of Salem, Massachusetts. That model apparently was built shortly after the ship's victory over the
Guerriere for presentation to Captain Hull. It's crude in lots of ways (if I remember correctly there are no trucks on the gun carriages, and each gun barrel is held to the deck by a big, conspicuous nail), but its rigging is extremely detailed. I think Mr. Campbell used that model as his source for the rigging on the Smithsonian plans. (The Humphreys drawing and the Admiralty draught don't show anything above the weather decks.)
Anybody with any interest in this subject needs to get hold of a copy of Howard I. Chapelle's classic book,
The History of the American Sailing Navy. Chapelle redrew both the drawings. (He also included, in his appendix, a contemporary letter from an officer who had served on board both the
Constitution and the
President. If memory serves, he said the latter actually had some superior sailing qualities. That letter writer also made a few observations that, from the modeler's standpoint, are a little disturbing. I think he said something to the effect that the
President's "quarterdeck is in my opinion longer.") I think the book is out of print at the moment, but hundreds of copies are available via the web - some at very reasonable prices.
Bottom line: except for the decorations on the transom, the Revell kit probably represents the
President just about as well as it represents the
Constitution. I'm not aware of any contemporary pictures that show the carved work of the
President in any detail. (To be honest, I haven't had occasion to look.) I think the Chapelle book also contains a set of spar dimensions for the
President; it would be worth comparing them with those of the kit. I suspect, though, that if a modeler were to change the name on the transom, nobody would be in a position to claim the result wasn't an accurate model of the
President.
Hope this helps a little. Good luck.