My interpretation is more strongly colored by the way Wolfman and Pérez originally developed Nightwing pre-Crisis, when there was no acrimonious break with Bruce/Batman. The Nightwing role was a sign of maturing, but not rebelling.
After the first Crisis the history of that relationship changed, but by then Nightwing was established with the Titans, and any conflict with Batman was secondary and in the past. The Nightwing magazine focused on Dick in his twenties and establishing himself as an adult.
I think the Red Hood, as used in this millennium, is a more obvious example of rebellion against a father figure and his values.