Time Capsule - S3-E3
Corrected entry: It's confirmed in the final season that Jerry (AKA Larry, Gary, and Terry) is actually called Gary. But in 3-3 "Time Capsule", he states that he would like to include his mother's book she wrote when he was little in the Time Capsule. The book refers to him as Gerald (Jerry) (00:00:30)
Correction: His real name is Gerald Gergich (shortened to Garry). Over the course of the show he has been called Jerry, Larry and Terry by various people and never corrected them out of politeness. His mother got his name correct.
In The Trial of Leslie Knope, he states his name is Gary and shows Leslie his license.