You cant seriously try to answer this without complex physics that deals with the phase transition mechanism and how it takes place across the body of water depending on the environment and the condition of the water that in principle is different from boiled and cold. The container geometry also may play a role. Also the exact content of water matters (is it 100% water molecules, eg real drinking water never is that plus it has minerals/salts etc).
I am also willing to claim the answer depends on how fast the system is forced to freeze.
I sure hope its not a hoax as i have no personal experimental familiarity with such effect so i take it now for granted by the references. It should be a physics no chemistry problem though unless its a physical chemistry subject that chemists are more advanced in dealing than many-body/liquid state/phase transition physicists. But real material scientists are the only ones that can do it right (ie physicists) if forced to do the real problem in all its glory.
Another thing to ask is if the ice formed is identical in structure in both waters.
Something i have observed however that is different but maybe similar in explanation ultimately is that water that has been boiled and then cooled and forced to be reboiled again arrives at the rapid bubble phase without the gradual bubble phase that the originally cold water had (just get water from faucet and boil it and observe how it arrives to rapid bubbling gradually, then coolit and repeat boiling, there is always an earlier gradual slower bubbling phase before the intense one sets in in the fresh water. Of course that could be a result of the air diluted in the water having been mostly released from the original boiling.
The very first experiment i would do is take water that has been thoroughly boiled many times to remove any air and then start the experiment with this type of totally clean 100% , air free water from same cold temperature always of course and then boil and try to freeze etc . Then report and only if then we still have a difference the problem is real interesting.
Also i am ultra confident the starting temperature of both waters matters and seriously you cant expect that water that is real cold vs 90 C degree water will still lead to 90 freezing faster. The hot water seriously takes time to become cold before freezing is possible. To bypass that extra time would be ridiculous unless they both start the attempt to freeze from close enough temperatures or the coldest one is not very close to 0 anyway.
Edit: Read some of what i said (concerns for set up) and other here http://en.wikipedia.org/wiki/Mpemba_effect