Right. Effectively, we're projecting every horizontal slice of the sphere laterally to a fixed distance from the vertical axis. It's a stack of 1D projections, not a single 2D projection.
I realized after the fact that the more useful terms would be “orthogonal projection” and “perspective projection”. The novelty of the orthogonal projection at hand is that it’s projecting (flattening) in a cylindrical space.
With an orthogonal projection, you can usually think of it as taking two planes and squashing whatever object your want to project between them. In the scenario here, the ambient space has been wrapped up, so one of these squashing planes has been wrapped into a cylinder, and the other has been wrapped into a line (the degenerate case).
In either event, an orthogonal projection is indeed a collection of orthogonal projections of one dimension less. But that’s not really the whole picture.
