I don't think so... The director may want to include scenes or change dialog but due to trying to get the movies rating kept to a certain level etc the directory is told to change stuff. At least that is my understanding.

The desire to keep a movie under a certain running time is also a major factor. The shorter a movie, the more playings (and thus, the more potential tickets sold) per day.

Peter Jackson intended for the LotR trilogy to have extended director's cuts. The theatrical versions were already well over 2 hours, but I am glad he shot the extra footage and kept it!

It depends upon the director's contract. Almost all of them don't get control of the final cut that makes the first theatrical release. Usually the production company gets the final say.

Besides with DVD sales they can save some to give viewers a good reason to buy it after seeing it in the theater.

All but a select few directors on a select few projects (think Scorsese on Godfather II) have much less control than it might seem. Most of the control of a movie usually belongs to the producer.producers--they generally have final say over most aspects, including things like run time, budgeting decisions, marketing, and have a heavy influence in the editing process. Directors, generally speaking, handle day to day stuff, the dirty work if you will, while producers handle the big picture stuff and intervene when there are conflicts or complications.

So to get to the point, a directors cut would be when the director has an altered vision of the movie, that wasn't released theatrically due usually to conflicts with the producer(s) or run time issues.

Most movies are actually an amalgam of "cuts," the writers, the directors, the actors, and then the producers job is to work with the editor(s) to make it cohesive and fit together. The only movies you see that are truly directors cuts are either when the director is also the producer and/or editor and/or writer (Cohen brothers movies are like that, and some Tarantino movies, as examples), or when a director is so big and influential at the time he can't be denied it (again, Scorsese with Godfather II comes to mind). The first is much more common than the second however.