Because of this I can't use a normal door closer like the one in the first drawing. Do you guys know of any door closer that can be used in a door installed like this?
If you have normal or near normal hinges, you may be able to use a hinge mounted door closer. They exist in many forms, with published ratings for doors weighing from 30 pounds to as much as 150 pounds.
One model does not require a hinge installation and appears to be mountable on the door frame.
The photo shows the heavy capacity version. This particular unit is found on Amazon (US) for a rather reasonable price, considering the capacity is listed at 150 pounds.
The above site also shows an installation video of about four minutes. Disregarding editing, it seems likely one could install this closer in four minutes.
Adjustable tension, various colors, seems like a great product.
If this doesn't meet your requirements, the search terms "hinge mounted door closer" should find something that will.