It seems to be a .32 Winchester self-loading made for the Winchester model 05, a self-loading rifle, introduced in 1905 and discontinued in 1920. It might have been the prototype or forerunner of the .30 carbine (not the rifle but the cartridge!). My source considers this cartridge as not very popular and obsolete.
Looking to your picture, it seems to me a reload.
The round is a late Peters style of headstamp after Remington took them over. You might try looking through / at the catalogs on this site to see when they stopped offering the .32 WSL.
I have exactly the same one & am sure it is factory.
The last PETERS catalog showing the .32 Self-Loading (.32 Winchester Self-Loading) cartridge is that of c.1951. It is an undated catalog, but has basically the same content as the dated 1950 catalog, and while in a different color scheme, a cover identical to the 1950 catalog, so I am assuming it is from 1951. The 1952 catalog does not contain this cartridge.
The headstamp style and flat nickel primer cup would indicate to me that the cartridge shown is from the last years of PETERS production of this caliber. It is a 165 grain, flat-nose, soft-nose bullet, the only load shown in the last catalogs (I did not check earlier ones to see if there ever was a different PETERS loading for the .32 SL).