The RM (S&W called it simply "The 357 Magnum") got its name from being "registered" to the original buyer. Many registration cards were never returned to S&W and they are now valuable collector items.
The guns were total custom jobs with buyers specifying barrel length (in 1/4" increments), stocks and sights. The registration program ran 1935-1939 then was dropped. The gun kept being made as pre-war magnum, post-war magnum, then in 1957 it became the Model 27.
Many if not most RMs went to police. I have one inscribed for the Utah Highway Patrol. It is one of 50 they bought (mine is #27). The S&W historian doesn't count these when he figures the number of RMs that were factory inscribed and I'm not sure why he doesn't include them.