My guess is the truth is probably somewhere in between.
The US isn't denying losing a drone in "Western Afghanistan" and it wouldn't shock me to learn it was actually over Iran. I doubt the Iranians shot it down, though. If, as they claim, it's "minimally damaged" then it probably just went out of control and crashed.
Looking at the map, if there's any truth that it was recovered near Qom, then it probably flew from Turkmenistan and never went anywhere near Afghanistan. I have no idea what sort of range these drones have, but Qom is about 500 miles from Afghanistan but less than half as far from Turkmenistan.
<edit> Having looked at the rather slim amount of available information about the RQ-170 on Wikipedia, it's much larger than I had imagined - roughly estimated 20m wingspan. Hard to see something that big crashing with "minimal damage".