Directors cut does not equel "All footage shot". it just means that it's the way the director intended it to be. In this day and age, "DIRECTORS CUT" really has been over used in many ways and gives folks a false sense of security that what they are buying is something more then they originally saw or something. Most directors are usually pleased with the film released in theaters.
as for the crow, I believe it is indeed what the director intended even though some footage ismissing.. maybe the director didn't want it in..