Musculoskeletal-ly speaking, I would just like to get a take from peers here who do what I'm calling "full-rest" pull-ups, where you pull up with arms straight above you, then go all the way back to a straight-arm position for each pull-up. Guys here in the gym say they can do so many pull-ups, but they don't go all the way. Anyway, I was wondering since most people do pull-ups the short way, in bursts with no straightening of the arms to the "full-rest" position, is that because doing it the long way causes ball-and-socket damage in the long term, or are they just doing it that way because it's easier. Which is better?
What you're describing is, I believe, called a "Deadhang Pull-up," in which you return to the Deadhang position at the end of the eccentric portion of the exercise. Simply put, this is what I consider to be a correct pull-up. That being said, your mention of ball-and-socket joint issues is not incorrect.
Whether you're performing a deadhang pull-up or you're just doing deadhangs, a critical component of the form is to ensure that you're "packing" your shoulders. That is, you should be retracting your scapula. This video shows how to properly deadhang and this video gives a lot of good advice on pull-ups in general.
If you're not locking out, I'm not counting it. They'd get 0 reps counted in a competition, so they're just fooling themselves into thinking they're doing lots of reps.
Additionally, getting a full stretch on the muscle (in this case, the latissimus dorsi and bicep in particular) before starting the rep will give you more of a payout if you're doing it for the actual training, and not just to show off.
Their delusions will hold them back, gains-wise.
If you're having some sort of internal competition, impose a lock-out rule. If you don't, people will just go down as far as it pleases them to get a higher count, and people will stop at different points. A lock-out rule will keep everyone honest.
Your question is basically one that addresses full range of motion (ROM) versus “cheat” movements. Exercising a muscle (any muscle) through its full ROM tends to provide more long term benefits than “cheating” for the simple fact that full ROM tends to correlate with strict form. Full range movements also tend to enforce a longer time under tension. Lastly, those that use “cheat” movements are typically more interested in the “numbers” (sets, reps, etc.) rather than doing the exercise(s) correctly.
In my opinion, full ROM movements win out every time.