All this great information and not a single photograph!
S-prefix MP .38 Special. I've owned several over the years. This particular one I've not even shot yet. I have a second MP that was made 1923 and was police gun. The barrel is loose on that one.
Colt Official Police .38 Special. Mfg 1929. I was told it was an armored car guard gun. It has a nice smooth action but.... The cylinder lock-up is loose with side-to-side play. In looking at it, taking it apart, it appears to be the actual cylinder locking pin where it fits into the recess on the extractor star. I've thought how it could be repaired and seems it needs two procedures. One is to make a scrapping tool to clean out and freshen the extractor recess and then modify the locking pin with a larger pin. Seemed like a lot of work for a very small issue but there it is. The Colt came with the Pachmayr Presentation and they feel very good.
The 1939 Stoeger's catalog lists the ammo.