The little .31's were Colt's bigest sellers for civilian guns. I've never heard them referred to as "police" guns but no doubt lots of law enforcement types carried them because they were small and light. The Colt referred to as the "metropolitan police model" was the 1862. It was a five shot .36 caliber on the same frame size as the .31's and was quite popular. Lots of them were converted to cartridge guns in .38 centerfire or rimfire.
The Pietta gun which they call an "1862 police" is NOT. It is just a regular 1861, six shot, .36 caliber with a fluted cylinder to resemble the 1862 police but is a much larger and heavier gun. I've taken Cabelas to task for advertising it that way but they still do it.
Uberti does build a true replica of the 1862, with the round barrel and fluted cylinder it is called the "police model", with the octagon barrel and engraved cylinder it is called the "pocket navy", both are .36 caliber, five shooters.