What that whole essay seems to be saying is that the hot water freezing faster is an illusion, caused by other factors which act upon the water.
Apparently, quite a few things need to be controlled in order to test the hypothesis that hot water freezes faster. The water must be degassed (to control for the change in freezing point caused by solutes). The water must be prevented from evaporating, which causes other complications—hot water can only be prevented from evaporating by putting it in a container that does not expand, and preventing it from expanding increases the pressure, and increasing the pressure increases the freezing point. So the cold water would have to be subjected to the same pressure. The water must be cooled slowly, so as to avoid the supercooling effect. And the water should be stirred to maintain an even temperature—but the stirring will lower the freezing point—although this may not be necessary if the cooling is sufficiently slow to prevent supercooling.
I remember doing crystallization studies back in undergrad chemistry classes. We had to put crystals into a tiny glass tube, and put the tube into a heater that took forever to heat. Then we sat there, looking through a magnifier at the crystal until it started to melt, and record the temperature. As I recall, the temperature change had to be very slow to get an accurate reading.
I also recall from those chemistry classes that a pure substance has to spend some time at a particular internal temperature before it undergoes a phase change. That is because the substance must gain or lose internal kinetic energy (depending on the direction of the phase change) before it can undergo the transition. Once the energy is gained or lost, the phase change occurs fairly rapidly.
I once heard on a science radio show that evaporation (aka, less mass to freeze) is the key to the story.