I do the same thing - there is a lot of diminishing return, set-to-set for me so it’s the only way that makes sense. I do inverted rows, dead hangs and arm circles to warm them up.
I guess this is more straightforward when doing weighted pull-ups. For bodyweight lifts I just start with my first set, no warm up. Maybe I'm wrong to do that, but it hasn't injured me yet.
I'm not sure I understand, but if your biggest set is your warm up I wouldn't call it warm up, then it's a working set. I never really warm up for pull ups since it's never my first exercise and I'm not worried about injuring myself doing them. But when I do warm up I only do like two reps between the last few sets of the exercise before.
Probably doesn't matter much but I prefer not to max out on my first set. If I were you I would do something like 7, 7, 7, 7 and start working toward 8, 8, 8, 8 when I can do four sets of 7. If you feel like you got a set left in you after that then go for it.
Personally I like to change things up with my pull ups every workout. Sometimes I just do two amraps, sometimes i pause at the bottom and sometimes I barely go parallel. But I generally leave a few reps in the tank the first sets.