their description and their graphics describe/show an equirectangular projection.
that’s not a cylindrical projection. they made a mistake in labeling it as such.
I won’t even try to understand those equations you presented, or why you’d even consider projecting onto polyhedra. I see no reason at all to complicate things to that degree.
you aren’t trying to emulate an actual physical projection. equirectangular is physically impossible but mathematically very simple.
you have regular pictures and you want to calculate things on them. that is the whole goal.
you know those 360 degree panoramic pictures, as seen on google maps and elsewhere? they’re almost always made from equirectangular data. the devices that produce these pictures consist of multiple cameras, and software maps each view into an equirectangular whole.