sometimes passive hanging without engaging ur shoulders isnt good cause the strain go to the tendons and not the muscles.
So especially if u are injured u should try to retracked ur shoulder blades abd push them down a bit and see if it feels better.
As for the warm up the vid is good. U can just do joing circles, small big , maybe some work with elastic bands for rotator cuff mostly.
Finally jogging etc to pump up blood and get the temperature high is good, and also doing an easier version of an easy set of what u are going to do is a nice way as well.
Hope it helps, have a fast healing btw
Goals: Be able to do bar-barians requirements in less than 10 mins, 10 good form one hand push ups, 15 pistols, Be able to do one hand holds on bar and controlled negatives, 1 min hanging l-sit, backlever hold for 5+ secs, advanced tucked FL for 5+ secs, advanced tucked planche for 8+secs etc etc
Long Goal (till summer): My 1st OAC,good BL hold, FL hold, bent arm planche, free handstand push ups, l-sit to handstand on handles, v-sit hold on floor