The basic rule of thumb for breathing is to breath out on the hardest part of the exercise (i.e. breathing out when pressing a
bench press up & breathing in on the way down)... So w/ push-ups, breath in on the way down and breath out going up. And for pull-ups you are doing it right...
The reason for this is when you hold your breath or taking a breath, it could decrease your range of motion by tightening up your muscles... Just think of how tennis players and
martial arts fighters breath... Both "yell" or breath out hard with they go to hit the ball or someone/thing... I don't know if this is 100%, but this is the stuff that "I heard through the grapevine"...