Can you describe "half-down pullups"? Do you just mean pulling yourself up from a half-hanging position, rather than hanging completely from the bar before beginning the pullup?
Either half, though I usually do the top halfs, which works the smaller muscle groups that you really want to build. If you watch people struggling for that last pullup, they usually fail in the top half. The point is that a single pull-up is a tremendous amount of work through a large range of motion, so reps are tough to come by. Doing halves lets you get more reps. 16 half-ups is better than 6 full-ups.
