Most were also made in 38 Special. Would the different cylinder sizes make a difference at all? The Special is a bit longer.
Yes, the Police Positive's frame is a tad bit shorter than the Police Positive Special or Detective Special. But I don't think it will make a difference. Likewise, the slight difference in cylinder dimensions shouldn't make a difference either. It might if it were the other way around...trying to fit a slightly longer frame into a holster made for a shorter frame.
My objective was to steer you away from the Smith J and K frames, because those fits will not do even if a holster you find for those is simply a generic scabbard and not actually built or molded to fit the gun well. Holsters for the J/K frames are common enough in stores where it should be easy enough to try the fit in find out for yourself.
I went to eBay and found a number of generic looking Detective Special holsters with thumb breaks at various price ranges including inexpensive ones in leather. Some of these might do well enough for cheap. Many of the older designs with top straps, like the the Hunter for example, are merely leather envelopes that do not exhibit the higher degree of fit that many of the more modern holsters do. That said, your gun will go into those made for the DS and they will work if you find one that gives a half way decent degree of tight-enough fit. But I really doubt that you will find a mass produced holster made today that is specifically made for your PP. So if you want to go the mass produced route, find something you like made for a DS that is not highly molded and or made of too-stiff leather.
Given that the PP was dropped from the Colt line just after WW2, it will be difficult to find a maker that will have a mold to make a custom fit for the Police Positive. There is just no demand for them. The one exception that occurred to me might be Lefty Lewis. Among his designs are some old school ones by the legendary Chic Gaylord, as well as many of his own.
I went to Lewis's site to see if he makes holsters for the Police Positive (and not just for the PP Special or the DS) and it states that he does. So you could get one from him that should fit very well for around $100, which is reasonable for a custom build holster of quality. If it were me, that's what I would do if I were serious about carrying your particular firearm in a well fittied holster. Such a holster would not necessarily need a thumb break or an over-strap to provide good, safe retention.
https://www.bellcharteroakholsters.com
Go here to find out if yours is a first or second issue.
http://www.coltfever.com/Police_Positive.html