That is when Dumbledore realizes Harry is a “good” Tom Riddle. We don’t know when he realizes Harry is a horcrux.
EDIT: In fact, it’s almost certain that Dumbledore realizes that Harry is a horcrux before that scene, or at least suspects it. It doesn’t look like anything in that conversation in particular would make him realize that, and he clearly knows it by that point.
That is when Dumbledore realizes Harry is a “good” Tom Riddle. We don’t know when he realizes Harry is a horcrux.
EDIT: In fact, it’s almost certain that Dumbledore realizes that Harry is a horcrux before that scene, or at least suspects it. It doesn’t look like anything in that conversation in particular would make him realize that, and he clearly knows it by that point.