Final answer:
The person who is fully in charge of overseeing the execution of the show during performances is the production stage manager, who runs the show from rehearsals to the final performance, ensuring that all aspects of the production seamlessly come together.
Step-by-step explanation:
The individual fully in charge of overseeing the execution of the show during performances is the production stage manager. This person is responsible for running rehearsals, and, once the production opens, they take on the role of running the show. The stage manager is pivotal in ensuring that each aspect of the performance runs smoothly, from coordinating interactions between the cast and crew to calling cues during the show.
While the director is tasked with the staging and the creative concept of the show during rehearsals and makes important artistic decisions, it's the stage manager who takes the reins during the actual performances. After all the work put into rehearsals and technical adjustments, the director's job is concluded on opening night. At that point, the stage manager ensures the continuity and quality of the show for the duration of its run.