I don't know where this "rigid" model comes from, but I never encountered it in the early days. Everyone knew you couldn't just complete a phase in totality and then move on to the next; there had to be overlap and stepping back up to an earlier phase as you discovered new things.
I suspect the "rigid" variant is merely the hyperbole that waterfall has been reduced to since there are no more proponents of it left. Doesn't make the Royce variant useful, though.