I am of the opinion that is if first important to know that both movements are multiplanar.
The first being executed in both the sagittal plane as well as the frontal plane and the second being executed in the sagittal and transverse planes.
During the lunge we know that the anterior and posterior compartments of the thigh and leg as well as hip are being used as well as the medial and lateral for stabilzation.
Reaching the arms overhead in the frontal plane require the fitpro to ask him or herself what muscles are involved in raising the arms overhead in the frontal plane (aka known as humeral abduction).
As a consequence of the arms being planed over the head, one must consider the muscles involved in scapulohumeral rhythm in addition to the muscles of the shoulder girdle.
Same thing with the lunge with trunk rotation. Same approach, same thought process considering the muscles that function in the transverse plan.
In a nutshell, the personal trainer who decides to use this movement, should not (at least in my opinion) choose this movement without having a good understanding of planes of motion, axis of rotation in addition to having a firm understanding of the prime movers, synergists, neutralizers and stabilizers during specific movements.
This is simply the thought process I go through when I chose exercises.