That's super interesting. I had never heard that explanation for the term. It's very much plausible, but I always thought they were called "fake books" because they were knowingly created and distributed without the legal rights to the songs. I've never really heard musicians convey the notion that the performances from fake books are "fake performances" or that you're "faking your way through it." I would have expected that jazz musicians who do have the rights to songs would still use a very similar form of lead sheet, especially given the improvisational nature of most jazz. I doubt many people would gig with full transcriptions—those tend to only be used for intense study of notable performances.
I thought the idea is just that the lead sheets themselves are counterfeit, as in "a fake Rolex."