I did not find .44-40 listed in the rifle section of my manual.
One rarely does. In fact I went through several old Speer manual which didn't list .44-40 at all. Many older manuals, printed between the time it basically went obsolete and its revival due to cowboy action shooting simply don't list it, as it wasn't very popular at the time.
Despite the fact that it first appeared in the Winchester 73, it is classed as, and considered a pistol round these days. Things were different in the distant past, where the same round was a rifle or pistol round, if it came in both.
Today its a pistol class round and rifles chambering it are considered to be pistol caliber carbines, mostly...