Pectoralis muscles as push muscles vs pull muscles
The pectoralis major is not just an adductor and medial rotator of the arm. Its clavicular fibres also flex the arm at the glenohumeral joint. This is the action useful in performing push-ups and bench presses.
And yes, its adductor action also plays a role in pull-ups and climbing (along with that of the latissimus dorsi, another adductor).

References and further reading:

https://www.kenhub.com/en/library/anatomy/major-pectoralis-muscle

https://www.kenhub.com/en/library/anatomy/the-shoulder-joint

Romanes GJ. Cunningham's manual of practical anatomy. 15th ed. Vol 1, Upper and lower limbs. Oxford: Oxford University Press; 1986. (ELBS edition, 1989 reprint). 263 p.