"The Notebook" director and his 12-year-old girl, Barbie, were spotted at Jerry's Deli in Studio City Saturday ... just a day after an Oklahoma judge awarded Nick lawful custody of her.

We broke the story ... Cassavetes' ex, Heather Wahlquist, squared off with Nick in an Oklahoma courtroom Friday, asking the judge for an emergency order to keep Barbie in her custody. The judge rejected her request.

So Nick and Barbie look copacetic in California now, but the custody battle ain't over -- they're back in a California family law court January 18.