If it fits loose it is 9mm....if it is snug/tight or dosent fit at all it is 0.30 cal.I do not see any type of marks or stamps anywhere on the barrel. A No2 pencil just fits down the bore. Almost exact diameter.
From that link Mrerick had:
"Several posts mention the "pencil test". The fact is that a hex shaped (sharp edges) No. 2 pencil Will snugly fit down a .30 Luger caliber barrel. A round No. 2 pencil generally will not fit down the .30 Luger barrel.Both pencils will loosely fit down a 9mm Luger barrel."
My guess if it is not a loose fit, then your barrel is a 0.30 cal replacement barell added after WWI. Then the luger got a Germany export stamp and was imported into the USA for sale between the wars. Maybe in the 1920's.
Some other things to check. Your PX gun has wooden grips and the 1918 DWM has black grips that are either plastic or bakelite grips. I think I see characteristics that the black grips are real bakelite. This is important because authentic bakelite grips can be worth something by themselves. If these guns came from the same owner he may have switched the grips. The black may go with the PX gun and the wood with the DWM. Carefully remove the wood grips and see if they are numbered. Photos of interior side would be good.
On the same note. The trigger on the PX is numbered 91, which is the last two digits of the DWM's serial #. What is the number on the trigger of the DWM? Could the triggers be swapped like the grips?
The tool with your PX gun is East German.