I think what saltfish's article is saying is that, during the supercooled timeperiod, the water remains in the liquid state. This is what supercooling is. According to the graph, the supercool state for hot water is much shorter than for cold water, therefore the hot water will become a solid once equilibrium has been reached at the freezing point. It is also asserted that this is assisted by dissolved gases in hotter water, as nucleation will bring any supercooled substance out of the supercooled state very quickly.
So, at the end of the day, assuming nucleation is the same in both situations (or no nucleation at all,) Hot water
truly does become a solid faster.