Scandalous was funded by McPherson's church and (less publicly) by Gifford. Atomic, an open-ended, very expensively produced Off Broadway commercial production (quite a different animal from festival shows) was funded by the the authors' fortunes, inherited from the Australian discount clothing store founded by their father. Holler was, in significant part, funded by the Shakur fortune, under his mother's name.
But really, none of this is worth any kind of argument; we merely define the phrase "vanity project" differently. To me, it's more about amateurs who think that they're artists, spending huge sums on dressing up dreck. I can keep my definition, and you can keep yours, and everyone will be happy. Either way, the theatre professionals whom these artists hire to dress up their dreck get a healthy injection into their bank balances.
I would also differ from you in that I wouldn't call The Visit a vanity project at all, as it had been financed in workshop form for years by theatres who have a lot of experience and no special relationship to Kander, Ebb, or MacNally. And the Broadway production boasted a roster of experienced producers/investors who also had no special relationship to the writers.
I suppose one could hypothesize that Honeymoon in Vegasbecame a vanity project, as it was Danza's money that kept it running for much of its time on Broadway.
You can define the sky as green and the grass as blue and I'll be happy. But being funded by the church of a show's subject is not a vanity production. And investing money to exploit a property you own on Broadway is not vanity: if that were the case, Disney is the mother of all vanity projects. Regarding the amateur thing-everyone was an amateur once upon a time; I don't want to be in the business of condescending to amateurs: we desperately need them. And so many so-called professionals in this business don't seem able to do anything better than dress up dreck so I don't know how you can express that distinction with a straight face. Finally, re The Visit, McNally's husband produced it, and if you look at the producing team, it is largely a bunch of close friends of those involved who have more money to waste than I do.