The '47 thru '50 trucks had NO vent windows and the 'pull down' door handles.
1951 was the first year for vent windows.. but still had the old 'pull down' door handles.
1952 was when they changed the door handles to the 'push button' handles.. and the same doors were used on the '53 trucks.
The doors on '47 thru '53 trucks are interchangeable.. you can take doors of a '47 and slap 'em on a '53 with no modification.
So the truck you have with vent windows and push button door handles would be the 'correct' door for a '53.. and the truck you have with no vent windows and 'push down door handles' would probably be doors off a '47 thru '50 truck.
One thing you might want to look at on the truck with no vent windows and 'push down door handles' is the presence of a 'crotch cooler' on the drivers side.. this was a vent panel that opened by pulling a lever that forced air into the cab.
You can see an example of one in
This picture just below the drivers side mirror.
It's not uncommon for people to come across trucks that are titled as being one year, but actually being another model year.
It happens when a guy has a good title and VIN plate for, say a '53.. and has a good '50 truck with no title. Swapping VIN plates gives 'em a good truck with a good title. Certainly not legal, but it happens.. and your average DMV dude isn't going to know enough about the trucks to tell the difference between a '50 and a '53.
More than likely in your case, what happened was somebody had a good '53 cab and needed some good doors, and scored a set off a '47 thru '50.