45 Government is the official name of that cartridge. It's specifications are in the manuals of the day. As mentioned it was made by government arsenals so the same cartridge could be used in both the Schofield and Colt service revolvers. Colt then began making the Colt SAA's with smaller diameter ratchets on the cylinders so regular 45 Scofield cartridges could be used. Many of the original issued SAAs were recalled and modified. The 45 Government cartridge had a short life.
45 Short Colt is a colloquial name.