Any .32 Long I-frame with serial number above about 258000 would be a Third Model, so you are correct. That gun probably dates to 1919 or 1920.
Does it by an chance have a square butt configuration whose wooden stocks overlap the bottom of the grip frame? If so, it would be called a .32 Regulation Police. If the butt has small wooden round butt stocks that leave the bottom of the grip frame exposed, and if the serial number is on the exposed bottom portion, this would be identified as a .32 Hand Ejector, Third Model. The serial number on an RP would be stamped on the forestrap of the grip frame behind the trigger guard.
The only different between the RP, which was introduced in 1917, and the .32 HE is the shape and size of the grip.