In the early '90s, somewhere between 1992-94, four new to circulation '77 audience tapes surfaced. Pretty sure Jacksonville was around before that, but Milwaukee was one of them, Richfield/Cleveland with Ronnie Spector and two others that are escaping me. At the time the story was these were, as you say, official audience tapes, done by the crew, from the board set-up area, but with open mics.
The sound on those four recordings certainly suggested the story could be true: little to no close audience sounds, no mic movement to speak of nor changing sound image, etc.
So no, I don't think they are fan recordings, especially given that they all surfaced at exactly the same moment. I can't say with certainty the story holds, but the physical evidence suggests it might be/could be true.
The last part of your post though, doesn't hold water. Cost has nothing to do with how you record. The recordings are the same, in that the board tapes we have from '76 and '78 were just two tracks to reel or cassette depending on the show, so it has nothing to do with the cost of blanks. What I recall being told is it was a choice at the time. Perhaps like the Dead, they were experimenting with the best way to document the shows (not to preserve for release) so at the time, someone thought audience tapes might be the way to go.