Part of the reason for confusion is because in 1896 Colt introduced a "New Police" revolver, designed in particular for the New York City Police Department. Most were chambered in .32 Long Colt, but as the .32 S&W Long became popular they offered that option too.
But of course they didn't want to put anything on a Colt product that had "S&W" in it, so they got the ammunition companies to bring out an identical cartridge (except for a small flat on the front of the bullet) and called it the .32 Colt New Police (cartridge). About 1908 they discontinued the New Police revolver and replaced it with the Police Positive, but the .32 Colt New Police cartridge continued on until shortly after World War Two.
You can easily tell the difference. Those that were chambered in .32 Long Colt had the chambers bored straight through, as is done today with .22 LR revolver chambers. Those that are chambered to use the Colt New Police and .32 S&W Long cartridges (which are interchangeable) have a step in the front of the chamber, as is the practice today in C.F. revolvers.