This question refers to pull-ups mostly, as I tend to always do a bigger number of sets of pull-ups, hoping it will increase my pulling strength. Is this, for example (where each number is a number of reps in one set)
9/6/4/3/3/4/3/2/3/3/4/2
in any way better than this?
9/6/4/3
Volume is an independent variable that can be used to overcome the body's acclimatization to training stimulus. This is particularly useful when the exercise is done for high reps, such as with simple bodyweight work like chin-ups, pull-ups, push-ups, and dips. (More complex bodyweight work would mean gymnastic variations to increase the difficulty of the move, usually through disadvantageous leverage.)
For example, I've found that pull-ups of 15/14/12 would stop doing anything. Increasing volume through extra sets, e.g. 15/14/12/8/8/8 or 12/12/12/10/10/10/8/5 or 1/2/3/4/5/6/7/8/9/10/9/8/7/6/5/4/3/2/1, helped build muscle.
With respect to the example that you provided, the additional sets might benefit you in subsequent attempts at longer sets (for example, 10/8/7), if you recover and adapt from the additional sets. The additional sets probably will increase your recovery demand, and the additional fatigue may negatively impact other exercises that require those muscles.
