"That'll Be The Day" was released as by The Crickets for legal purposes. Holly was under contract to Decca Records at the time. But he didn't get along with Decca producer Owen Bradley and disliked the way his records there sounded. So Holly, Jerry, Allison, Joe Mauldin and Niki Sullivan went to Norman Petty's Clovis, N.M. studio to record a demo of "That'll Be The Day."
Petty became the band's manager and sent the demo to Brunswick Records in New York. Brunswick released it under the backing group's name, which Allison came up with.
Brunswick Records and also Coral Records were subsidiaries of Decca, and crediting records separately became a legal loophole.
That's why only Buddy Holly's name appears on Coral releases; The Cricketts only on Brunswick releases; and the full name of Buddy Holly and the Cricketts on Decca releases.