I think the point was to contrast those fisheyes which produce a circular image on the sensor, from those whose image covers the entire sensor. It's true that this has nothing to do with the type of projection, so the sentence could be clearer.
In fact there are two distinct issues: size of image (does it cover the whole sensor, or just reach the top & bottom edges); and type of projection. The Samyangs (the 8mm works the same on APC as the 12mm on FF) have a different projection than eg Canon fish-eyes, so they seem to bulge less in the middle. That's probably why the de-fish seems to work rather well.
There are other types of projection: equi-distant, equi-solid-angle, gnomic, etc etc. See under "mapping function" halfway down:
https://en.wikipedia.org/wiki/Fisheye_lensBTW, there is an error: the article claims 180°
horizontal angle of view. If that were the case, there could be no sides on the hood. I think the 180° is the diagonal angle of view, as is the standard measure for all other lenses.