to oversee a task or group, taking charge so the work is carried out well
At its core this verb means to be set over something and see it through: to take the lead, supervise, and keep a job moving toward completion. The Chronicler and Ezra use it for the people put in charge of building and running the temple, the foremen who directed the laborers and the Levites who superintended the service.
The same idea of taking the lead produced the word's most famous use. The form that opens more than fifty Psalms, traditionally rendered "to the choirmaster" or "to the music director," names the person who led the temple singers and players. It is the leader of the music, the one over the performance, which is simply this oversight applied to worship.
So the supervisory use and the musical heading are not two separate words but one: someone is placed over a task and made responsible for it. The setting changes (a building project, the Levites' duties, a choir), but the role is the same.