Just to give you some additional caliber information ( and confusion!)
S&W developed the 32 S&W in the 1870’s for the small top break revolvers, then lengthened it for use in the small I frame hand ejectors. It would later be lengthened twice more, once to make the 32 H&R magnum, then again to create the 327 federal magnum. ( actual bullet diameter is about .312 so really a 31 not a 32.)
Colt started with the 32 colt ( short case) then lengthened that to a longer case in a similar fashion. The key difference is unlike most centerfires these used an Outside lubricated bullet close to the same diameter of the case ( much like what you see in a 22 long rifle) with larger bullet/ bore diameters
This was phased out in the early 1900’s over time.
Colt didn’t want to put 32 S&W marking on their guns, so designed a “new” 32 cartridge called the 32 colt new police ( often marked on guns as 32 NP) which is simply the same 32 S&W long case with a flat point instead of a traditional round nose bullet.
32/20 was also called 32 Winchester or 32 WCF ( Winchester centerfire ) and was chambered in the K size S&W and colt ‘s similar size frame as it was too long for the smaller frame sizes of the early 1900’s. When using this cartridge, caution must be advised. The ammo produced in the last 30-40 years in 32/20 has been at a pressure level safe for both rifles and pistols. Back in the 1950’s and earlier, to provide more power in rifles like the Winchester model 1892 which is a strong action some ammo companies made a high pressure/ velocity load designed for rifles only which is far too high pressure to use in a revolver safely. These are often headstamped 32/20 HV M92 or similar.
I used a lot of 32/20 in both a rifle and pistol years ago, and any random loose old ammo I found was always fired in the rifle first, then used for reloading to a pressure level safe for both types of guns.
32 colt or long colt I don’t think has been factory made since the 1960’s, so all you will find for that will be very old stock ammo. Old stock is not at all bad, but you will probably have to visit some crusty old gun shops to find it