There are 4 primary considerations that must be met if you want the cylinder to fit and work properly.
Over the years, I have fitted many Ruger convertible cylinders. Regardless it sounds like a nice shooter and you can always simply enjoy it for what it is. Given the info we have so far it could have shipped as a either a 22LR only, a 22 Mag only or a convertible. If it really matters to you contact Ruger Customer Service and see if they can look up your serial number and confirm how your Single-Six originally shipped. 22 CAL SINGLE-SIX' instead of 'RUGER SINGLE-SIX WIN.22 RF MAG CAL'. Your serial number does fall within the 1962 year and the range where some magnum only guns were shipped and rollmarked 'RUGER. Anybody's guess how you ended up with a different cylinder at this point. Or it was shipped as a magnum only version or 22LR version originally and a different cylinder was found at some point for some reason. Perhaps the original magnum cylinder was lost to a convertible and a replacement was found and now the 22LR cylinder is lost. Since the last three digits scratched onto the cylinder do not match the frame serial number there are a couple of possibilities that come to mind.