What is the best method for painting a door
Neither!  Your door should be painted the same color as the trim, which is usually in a glossier finish than the walls. Semi gloss or gloss is typical for trim and doors.

The best finish is achieved when you remove the door, lay it flat and paint it with a good soft bristle brush. If it is a raised panel style door, start by painting the inner most details first, inner stiles second and finish with the outer stiles. Finish with long strokes going in the same direction as the grain.  Painting the door while laying flat will help eleminate runs and sags in detail corners.  Good Luck.