I think a lot of this confusion stems from the author framing this project in a way that is, frankly, nonsense. Vanishing points have nothing to do with the clipping plane, and 3D computer graphics isn't trying to approximate the coordinate transform described in the post, it's trying to emulate real-life optics. The sense in which this definition is "strict" is based on ignoring that N-point perspective is just a tool for drawing lines parallel to the coordinate axes with correct perspective. If the author had framed this as just a fun "what if", making every line converge to the vanishing point, it might have been better received.

The observation that some lines look strange is about as meaningful as the same observation on an any given map projection. It's definitely true, but it all depends on the choice of mapping function, nothing inherent to spherical geometry.

I choose map projections for a particular reason: as has already been pointing out, there's no connection between this "strict" definition and an exponential coordinate transform. Any function mapping the axes that grows faster than polynomial would produce the desired behavior (essentially, you need the ratio between z and x on a line a*x + b*z = c to approach zero or infinity), but would produce different "line" shapes. This choice is entirely arbitrary, and is in fact equivalent to simply mapping the underlying space by a related transformation and then using a normal projective transformation.