The chest (pecs) help to keep the upper body stable while doing chins/pull-ups. So they are brought into play/worked, being the opposite muscle group of the back/lat's. The pecs muscle action is brought into it to a lesser degree. An example of a reverse muscle action is that the lats are brought into play when benching, as a support factor. The abs are also well works when chinning. And of course the biceps. Which just shows that the body does act as a unit.
Chin's are pretty much just that, to or under the chin. Pull-ups tend to be a motion where the upper pecs/chest are touch, each rep, with the bar at the top position. This would be Gironda and other BB'ers of the '60's-70's style. More of a complete motion. Don't see too many guy's doing that version except on a lat machine. The names have become pretty much interchangeable throughout the years. Than there are pull-up, where the back of the neck is touched, each rep, to the bar. Gives a newer slant to the posterior delts and the lat's themselves.
Hand grip, under or overhand, are usually interchangeable with chins or pull-ups. If your gym has a hammer grip on the chin/pull-up bar than you may find that more agreeable for doing any version of the exercise. If the gym has a set of gymnastic rings, than you are in luck. They allow the natural twisting of the wrist and elbows, giving less tension on the shoulder joints themselves. And may offer a more complete form for any chin/pull-up exercise. Also affects the biceps greatly. If in doubt, check out the lat & arm development of most any gymnast.
Try adding extra weight (weight belt) to a chin/pull-up workout. Can be on a equal par with heavy BB row as a superior mass builder. Good Luck.