Dragoons, as originally conceived in the late 16th or early 17th century, were primarily mounted infantry riding to the battle and then dismounting and fighting on foot with fighting from horseback a very secondary consideration, the idea being I guess to combine the mobility of cavalry with the firepower and staying power of infantry. The Battle of Naseby in 1645 during the English Civil War offers what looks to me like an example of how dragoons were intended to function - the Parliamentarian dragoons initially deployed on foot behind a hedge and did significant damage with their muskets, later mounting and charging the Royalist flank when the opportunity presented itself.
By the middle of the 18th century, though, dragoons had evolved into a general-purpose medium cavalry type, armed with saber, a pistol or two, and a carbine. This may have been because cavalry was a more prestigious arm than infantry, so national and unit pride encouraged mission creep towards true cavalry status (particularly since dragoons were already almost as expensive as cavalry to equip, despite poorer quality mounts and swords). It may also have been the case that the hybrid units, while good in theory, tended to combined the disadvantages of both branches, being more expensive than infantry yet offering less firepower since every fourth or fifth man was tasked with holding the horses at the rear instead of being in the ranks firing his musket, while dragoons could not maneuver and fight hand-to-hand as well as true cavalry (having to train for two tasks instead of one, plus poorer quality equipment, I think.) While, contra certain modern authors, it is possible to find examples of mounted infantry that did well (Okey's dragoons at Naseby, Johnson's Mounted Kentucky Rifles in the War of 1812, plus most of the patriot militia in the Carolinas during the Revolutionary War rode horses and dismounted to fight), it is plausible that there weren't very many roles that a hybrid unit could fill that couldn't be done as well or better than a standard infantry or cavalry unit.
So by 1798 a dragoon unit was just a cavalry unit, more specifically one that could do duty as a light cavalryman (scouting, skirmishing, pursuit of a broken foe) and a heavy one (charging enemy cavalry and infantry). While European powers still had multiple differnt types of cavalry with different roles (hussars, uhlans, lancers, curassiers, etc.) British had already transitioned all their cavalry into dragoons (differentiated only into light and heavy dragoons) by this time, and the US never raised any other kinds of cavalry until mechanization.
While Neumann illustrates a French 1770-71 Dragoon musket with a 42" barrel, indicating that (if his ID is correct) they were still producing longer weapons for dragoons at that late date, cavalry carbines were generally considerably shorter and a 39" barrel would have been rather long for weapon intended for a true cavalryman. I wouldn't be at all surprised to learn that these short muskets were made as a copy of an older French weapon, but never adopted or produced in numbers because the role of dragoons in American service at the beginning of the 19th century differed too much from the role of French dragoons half a century before. Put in simpler terms, American cavalry needed a gun they could stow in a carbine boot on the saddle instead of being forced to sling over their shoulders (I also note that this gun has infantry style sling swivels mounted underneath instead of cavalry style mounted on the side of the gun.)
