I've never been through this myself, but I can't imagine someone honestly questioning whether or not Danny is short for Daniel and Don or Donny is short for Donald.
If the nicknames were other than very common "short for" very common formal names, I might be concerned. But these are very, very common nicknames for their given names.
But I'm with you. It drives me crazy when people don't realize that there is a time for formal names. My niece's name is Elizabeth, but we often call her Lizzie. But I know better than to write a check out to Lizzie. Yes, I realize it's done all the time, but it bugs me. Checks, flight reservations,etc., should carry the same name as a birth certificate.
ETA - And can I just add that I think it's ridiculous that someone - the airline, the agent, whoever, just someone - can't make a simple change in the computer. Seriously?!?