My general sens is that the term is very flexible and not completely nailed down to one stringent definition.
(I know. Not terribly helpful.)
Well, I acted as a "creative consultant" on the show "Monsters" for Laurel Entertainment.
Monsters didn't have a "show runner" in the traditional sense. In fact, there wasn't much about it and about the way it was run that was traditional.
It was an anthology show. There was no writing staff. Every script we ever had on that show was written by a freelance writer.
A number of those writers were people that had been involved with Laurel from the "Tales From the Darkside" days -- others were people who had submitted material to us that we'd bought.
Part of my work was to find material -- a certain percentage of episodes were adapted from other works, old and new, so it involved a lot of reading. The rest came from submitted materials - teleplays and treatments that came either through agents or over the transom. We were one of the few TV shows that were willing to read unagented submissions, if it came with a release form.
So I supervised interns who logged in, read, and covered that material. Anything that looked good, I'd set aside. For the rest, there were three possibilities. If it was outright crap, it had the form letter that said, Thanks for thinking of us. If it wasn't completely crap but just wasn't right, it said, Thanks, we'd be happy to look at additional material. Or, if it was good but for whatever reason (just like something we'd already done) I might write a personal letter saying in specific terms that they should submit more material and thanks.
As far as the stuff that fell into the realm of what might end up as possible episodes -- with the stories, I'd write out my own coverage -- really my take on how the stories might be developed as episodes, and also pass on the teleplays or treatments that I thought might be worth developing.
Then, with the V.P. and President of the company, they'd either approve or disapprove, if a project needed a writer we'd try to figure out who might be right for what script, whether something should be shot east coast or west coast (because we split production up that way).
Then, everyone would do notes, I'd collate those notes, send them to the writer, be the point person in those conversations -- just generally make sure that the various episodes were consistent with the physical requirements as well as the overall tone of the show.
But I couldn't begin to tell you whether or not that's what a "creative consultant" generally does.
NMS