> Actual waterfall looks very much like actual agile, in that it is designed around iteration loops.

No, Waterfall was not agile, it was the diagram from Royce but not what he recommended from his paper (which tore down that diagram). What Royce added to that diagram (fundamentally, just common sense with feedback loops) was closer to agile, though. Royce himself never called anything Waterfall, but what was termed Waterfall was the bad process he tore apart.