His birth name was actually Conway, which is the name they used for Burke in the movie.
Not true - The real name was Burke. They had to use Conway in the movie because of Burke's daughter trying to extort the studio to use the real name. In the end, they told her to go jump off a bridge and used Conway.
I believe Ivan is correct. Burke was his adopted name and when the producers couldn't use the name Burke they went to Jimmy's birth name.
