Yes, I think this is part of the reason the pet forms show up higher on these lists. There certainly are some people who get the pet forms on their birth certificates, but parents in the USA (as opposed to the UK) are much more likely to put the -y pet forms such as Jimmy, Donny, Billy, and Bobby on the birth certificate than they are to use Jim, Don, Bill, or Bob.
I think the name that this has affected the most in the earlier SSA lists is Joe. Joe is extremely high for birth cohorts before 1960 in the SSA data. Back when I was collecting data from state birth records, Joe was almost never used as an official given name. So I bet most of the Joes in the SSA data were actually named Joseph by their parents.