You must have missed Unit Ops, when they talked about heat transfer, how easily a materials allows the heat to move. Unfortunately the methods in school are pretty poor at really demonstrating this, at least in thermo. You needed unit ops to have seen it in school, as far as I know. Air holds less heat, thus it is a better insulator. Water holds more heat, thus it is an inferior insulator. What is the more significant factor, holding the heat, or transferring it? I still believe it is the transfer, if it weren't we'd be insulating with water as I think I'd mentioned before. No, water is what we use when we want to transfer heat better. Ultimately those experiments will be interesting to see the results. There is the convection in both spaces (whether it's air or water), the film effects (which favors transfer), so on and so forth.
This side of the argument assumes that since the ice lasts longer with the water drained (again, back to heat transfer since ice submerged in water melts faster), does it really keep the cooler colder longer? Does the loss of that cool mass adversely affect the cooler's air temperature? For me I'm trying to keep food cool, so air temperature is important. If you're cooling beer you may not care about the air as you'll put it in the water. I'm very interested to see his results.