Is there an equivalent of what in theatre is a stage manager for music? Someone that either stays at a specific venue, or travels with a band, and manages the light, sound, and other effects used in concerts?
Touring bands with a large budget and an elaborately produced show can have an entire production crew consisting of a production manager and a stage manager and a lighting manager, a video manager a sound manager and many more. The production manager would coordinate with the lighting manager and sound manager to be sure the "show" goes as planned. They communicate with a private frequency communication system wearing headsets and can constantly communicate with one another during the show.
High budget touring bands are not just concerned with optimal sound, but have a light show (perhaps even pyrotechnics) and many special effects as a means of creating a unique and exciting audience experience - beyond just the music itself.
Some touring bands may hire an independent sound crew and light crew who travel with the band and set up and tear down the sound system and light system and operate under the direction of a production manager.
Some bands will hire sound and light crews locally at each venue, and leave it up to the tour manager to coordinate hiring those trades in each venue.
A band with a lower budget and less elaborate production might have a production manager of sorts who supervises the stage set up, the sound reinforcement and the lighting and helps the band manage their sets to keep the show on schedule.
Small bar bands might just have one of the musicians who takes the lead and keeps everything on track and the other musicians on the same page.
There is not a regimented standard procedure that applies to all bands. It varies from act to act and is largely dictated by the budget.
Even if you are referring to just 'Theater' only, the range is very wide. When I worked in our small town (100 000 inh.) opera house there was a SM in the original sense of the word. Small equipment - for athmos or sound-effect playback on the fly. Light- and Sound-desk all backstage right behind the curtain. But later in the big cities I found impressive rigs. In Vienna where almost everything is aimed at Musicals and real commercial use even the old 'Raimund Theater' or 'Theater an der Wien' where they used to play Moliere and Nestroy are equipped monstrously these days. Light and sound all FOH - like a real Rock-Stage...
So I guess the same range is applicable to all different touring-scenarios... - The bigger the set the more control needed!