Howdy
According to Kuhnhausen's Colt Single Action Revolvers Handbook, a Shop Manual, the Army designation for the 45 Colt cartridge was Colt's Revolver Cartridge, Caliber .45, M1873. Here is a photo of a box of ammo I bought last year. In case you can't make it out, it was manufactured in 1874.
These are Benet primed rounds. They look like rimfire, but they are not, they are centerfire. As you can see, they have no headstamp. You will also notice these are the reduced power 30 grain loads, not 40 grain. They probably have cardboard wads inside them to take up the empty space, but I am not going to take one apart to find out. The round at the far right is a modern round for comparison. Yes, these rounds have tiny rims, and would never have functioned in a rifle.
According to Kuhnhausen the Schofield round was designated Revolver Ball Cartridge, Caliber .45, M1875. 1875 was the first year of production of the Schofield revolver. Unfortunately I do not have any antique Schofield rounds in my collection to photograph.
Here is a photo of a bunch of old 45 Colt rounds in my cartridge collection. The one all the way on the left is modern, for comparison. Most of these rounds have relatively small diameter rims. But notice the round all the way on the right. This one has an extra wide rim, about .540 in diameter, far larger than the current standard of .512. I believe this round was made for the Colt New Service double action revolver, adopted by the Army in 1909. The wide rim was so the extractor of the New Service revolver could extract the round.
Here is a photo of the headstamp of the round. FA stands for Frankford Arsenal, 12 13 is the date code meaning the round was manufactured in December of 1913.
I just checked, and the round does indeed chamber in my New Service revolver and the extractor grabs the wide rim.