You were unfortunately right the first time. I can walk through some of the reasoning for the sake of education:
The right wing on the reverse seems to be punched super deep, to the point that it has deformed the obverse.
If one side of a coin is badly damaged, the metal can be displaced on the other side. This can't really happen at the mint since the other side is backed by a die and the metal doesn't have anywhere to go.
The lettering and rim on the reverse seems to be completely messed up. Almost like off center, specifically around the "dollar" portion of quarter dollar. And the same seems to be true on the obverse side. Where the tail end of the "2" in 1972 seems to stretch into the rim of the coin.
The rim has been partially flattened (and possibly abraded) to the point of swallowing up some of the devices. It appears wider in places as a result, but the coin's design isn't offset. Other weirdness in the lettering and rim is just a result of metal being displaced from damage.
The entire quarter is not round, however the edge of the quarter still appears to have the vertical marks (around the edge) even though it isn't round at all.
The reeding (vertical marks) on the edge is imparted during striking by the collar, which also constrains the coin's size and shape. Any error that results in the coin not being round, like a planchet clip, also results in the metal not making contact with the collar in those deformed areas. The coin therefore lacks reeding in those areas. On a damaged coin, the existing rim is just pushed inward, leaving the reeding intact.
Also, under magnification (which I unfortunately can not get a clear photograph of there appears to be a very light 9 stamped perpendicular just under the "st" in trust on the obverse.
, I suspect. Our brains love to find patterns where there aren't any, and I see a lot of little stains and scratches in that area that could be interpreted in different ways.
I hope that helps. Keep looking and you'll develop an eye for this sort of thing.