Corrected entry: The crew manifest lists Dr. Crusher's rank as "Lieutenant Commander", but her 3 rank insignia show her to be a Commander.
Correction: MacDuff, the alien, changed some computer logs. He inserted himself as first officer and dropped Riker's position. He could have altered Crusher's rank too. (They had no idea what the rank insignias meant. That memory was erased.)
Correction: MacDuff had to alter the computer files in order to insert himself, demote Riker and add the information about the "Lysian War." It's possible that he ended up messing up some other files while he was at it, and no-one would have noticed anything wrong because their memories had been wiped.