Comet keith provides a great variable in the topic for discussion.
The variables at play when the pond is not running is aeration and gas exchange. Most people use aerator stones or rubber membrane defuses.
The question is what is happening to the water when these are used? Cold ambient air is injected into the pond water at the bottom of the pond where some is dissolved into the water while the rest exits the pond causing gas exchange.
I really do hope everyone uses aerators for both aeration and keeping a hole in the ice. if you do use a heater, please put it on an analog timer or just unplug it when it is warm.
dissolved oxygen is used for three things in a pond: Fish, Decay and plants. all three of these must have oxygen in the winter.
But the real variable is will the fish benifit more from 4 degree water or 2.3 degrees less with very healthy water. Lets take out the fact that aerating the water will drop the temperature just the same as running the water.
I will be honest, I have serviced over 300 ponds in Calgary and one thing I know for sure is that a lot of ponds need more than one air stone. while a pond can have nice decay around the air stone, across the pond through a narrow area (perhaps the pond narrows for a bridge) the pond may not be getting oxygen and have anaerobic decay (oxygenless decay).
a good way to test this on a smaller pond is the water in your skimmer in the spring when the pond is ready to start up. make a point of pumping the water out of the skimmer before you hook up the line. That water is almost always stagnant and smells like rotten eggs. Don't pump it into the clean aerated water.. oops I am off topic.
I don't have a lot of substantial evidence for this and I don't like to just assume anything, but I do think the fish do better with a healthier water over the temp difference. water quality is key.
I will be honest. pond builders up here build ponds with no filtration and tell the clients that a pond will filter themselves. Scary isn't it?