That's a real beauty!
You're correct, it's a .32 Hand Ejector built on the Improved I frame, the #568013 was most likely shipped 12/51. You have the 1st variation Improved I frame with round sight.
The 2nd variation from 1952 has the barrel rib and ramp front sight. Both variations have five screws with trigger guard screw. The four screw began with the Model of 1953 (the true Pre Model*). The screw count is different for I & J frames and start with the 6 screw for I frames (the post war transition I frame that preceded yours).
Explained in the new SCSW-4 ed., pg. 144 and here:
It's too early to be a Pre Model in the explicit "Pre-Model" collector term meaning used in the context of S&W collecting and should not be confused as the generic meaning of pre model which is a common misuse of the term.
“A Pre-Model is any ‘named’ model that immediately precedes when the company began the model ‘number’ stamping system, and incorporates all of the same design changes of their model numbered counterparts.”