Where are there what appear to be two different serials of differing length?
Can you tell me why the numbers are different on the various parts
The serial number is on the butt and on the back of the cylinder. The numbers inside the yoke are assembly numbers. It's pretty common for folks to confuse the two.
Also would this have been a police issue gun, or commercial?
Anything is possible with Smith. When your gun was built, the police market wasn't as sophisticated and specialized as it is today. Police guns were nearly always standard commercial models, but a few departments did order small custom runs. It wasn't profitable for Smith to do otherwise, and there was less need to standardize weapons because there wasn't an army of lawyers waiting to sue departments every time an officer fired a shot in the line of duty. Also, some departments let officers carry their personal weapons, so the whole question is really hard to pin down.
The best way to confirm departmental history is a factory letter stating that it was shipped to a specific department. Departmental markings are another way. But some department-issued guns weren't marked, and some that are marked were never issued and were later sold as surplus.
Military models differ from commercial models because they were built exactly as the ordering country's military wanted them and in far larger quantities. A typical military handgun order is in the tens of thousands, but in the 50's, only the biggest departments would have ordered more than a few dozen. After production for a military order was finished, Smith often varied the finish and markings, then offered those guns as a commercial model.
I realize there's a lot of gray area in the last three paragraphs, but like I said, anything is possible with Smith.
So long story short, your mother-in-law's revolver is certainly what a well-armed detective would have carried back then, but it probably wasn't built for that express purpose. Were there cops in her family?
Okie John