Cartridge names bear only a passing connection to the actual caliber. Basically the cartridge name is what ever the inventor or original manufacturer wants it to be.
Sometimes there is a historical connection, or some descriptive name, but it can simply be what the maker thought would sell best.
To answer specifics, .45 auto is the same as .45ACP. ACP stands for Automatic Colt Pistol.
.357 Magnum is NOT the same as .357 Sig. Other than the numbers (a reference to the caliber bullets fired) they are completely different cartridges, different length and shape.
A reloading manual or an illustrated catalogue from an ammo maker will give you a good comparative visual, and the reloading manuals often contain a brief history of the cartridge as well. I'm sure the info is also online, but I couldn't tell you where to find it, as I'm not well versed in what website has what....
All else being equal (and it almost never is) bigger bullets tend to work better.