Director's cut: A director's cut is an ordinary edited version of a film that is supposed to represent the director's own approved edit.

Cut explicitly refers to the process of film editing: the director's cut is preceded by the rough editor's cut and followed by the final cut meant for the public film release.

Fine Cut: The fine cut is the phase after the assembly edit and the rough cut [rough organiztion of scenes] stage. In this stage the music is finalised and an EDL is created to log the scenes. Overall, you eliminate any extraneous scenes or lines of dialogue and fix anything else that isn't working.