Go to page 37 in Jami Bernard’s book. It says that "Quentin only took back his father’s biological name of Tarantino because he couldn’t stand being called Zastoupil - in school, they had called him ‘Disaster-Pill’"
If he went by Zastoupil in school, that had to be his legal name.
And on page 17 in “Quentin Tarantino: The Cinema of Cool”:
“When Tarantino was aged two, Connie moved back to Los Angeles where she re-married, to Curt Zastoupil, a local musician. Curt adopted Quentin at age two and a half, giving him his surname. In fact, it was only when he had left school and decided to become an actor that Quentin Zastoupil reverted back to the more stage-friendly moniker of Tarantino, the name of his biological father.”