Nothing is actually correct or wrong. The best way to check soemthing is to try it urself!
its common in general to split the routines to one day upper body other day lower body and alternate these routines through the week.
Doing weighted basics isnt bad, there are planty of good routines out there that combines both BW and weighted basics.
One simple is 5-10 sets, of pull ups do max reps with lets say 15kg, drop off the weight try max again, rest 5sets between sets.
Other routines could be 4-6sets with weighted, 4-6 sets unweighted and u could do like 4-6 BW rows.
You could also do different routines for upper body. Lets say one day going a bit heavy, other day doing combined weighted-BW sets, other day pure BW basics.
Try to find out what u want to do/your goals and try to make a routine that will help you get there
Hope it helped.
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