I have heard that explanation, but I suspect that in more recent years the distinction was for .45 Colt vs .45 ACP.
Actually, there was never any ".45 Schofield" as distinct from the "combination" cartridge. When the Army adopted the Schofield, they adopted the cartridge also, calling it simply the ".45 Revolver cartridge." There was no confusion in service, since from that point it was the only .45 revolver cartridge issued as long as the .45 revolvers were in service.
As rcmodel says, the rim was the same as for the .45 Colt, but the case was shortened for use in the Schofield.
The round that made a "three shooter" out of the SAA was the Model 1909 cartridge. The Model 1909 swing-cylinder revolver (New Service) was made for .45 Colt, but the army found that the .45 Colt rim was too small for reliable extraction, so they made their own ammunition with a larger rim. That was the only ammo ever issued with that revolver, and it was never made except by Frankford Arsenal.
Jim