I have heard of a sound f/x person (different person from the one running the board) referred to as a sound coordinator. Also, we have on occasion used someone who provided various walkup music for an awards show. I called him a DJ but he would frown when I said that. Maybe he was a sound coordinator as well.
Coordinators tend to be the ones managing people, purchasing and paper work.
A designer literally does just that, they determine the best possible implementation for the equipment they have on a show by show basis.
This sometimes includes collaborating with a SM to work up a wireless microphone assignment sheet.
The experaince that i have, coming from a high school tech dept., is there is one person that ran everything/did everything for the sound : (. That always made it fun, we are about to produce monkey house, and i am really pushing for more people in the sound dept. But that i have found working with a local production company is; The Sound Coordinator is responsible for the design, implementation, and cueing of sound effects. And the Sound designer does ya just that; the best lay out and implementation of the equipment relative to the event that is being produced
From my experience, the sound designer would be primarily in charge of coming up with the concept and then working out a way to implement it. If the production was large enough, a coordinator would worry about the physical plant so the designer could concentrate on the concept. Think of the coordinator as roughly equivalent to a master electrician.
My guess is that the term is being incorrectly used. As has been posted on other threads, the common terms are "Sound Designer", "Sound Engineer", "Audio Engineer", "Sound Guy" (if you're male), "Sound Operator", "Sound Board Operator", "FOHMixer", etc. The difference within this list is that the first three terms generally imply the person who sets it all up and maybe also runs the show, the last three are the person who specifically doesn't set it up but does run the show, and the "Sound Guy" in the middle can be both.
Never heard of a "coordinator" position. At Ithaca, there is the sound designer who is responsible for providing sound cues, recorded music, vocal/orchestral reinforcement and the design of the reinforcement system. The sound engineer manages the sound crew and is responsible for the installation and integrity of the sound system, following the designer's specs. The mixer is responsible for the operation of the design. On reinforced shows this means learning the show and knowing pickups and ensuring adequate levels for audibility. Usually on shows that are not reinforced, we do not have a mixer position. On the larger shows, there may be an assistant sound engineer who may run the cue software/hardware during the show and assist the engineer with the management of the crew.