I'll argue that I have tried both ways.
Last June, we found ourselves without hot water from the tap. I would warm some water on the stove, but I found myself rinsing between passes with cold tap water (it was summer, it felt good).
I found that as time went by, I started pulling more hairs with cold water than if I kept the skin and whiskers soft with hot water.
Blades last longer with hot water, too. They are able to cut more effectively if the whisker is still hydrated. It was my experience that my blades gave rougher shaves sooner if used with cold water, and my theory was that the edges were getting beaten up by the uncut hairs.
I will say that I agree that you need to keep the skin hydrated between passes, and rinses are, IMHO and the experience of many others, vital for a successful shave.
Brushless or lather, boys, which ever you've the notion,
fifty cents for shaving cream, a dollar for the lotion; "Old Spice is quality," said the Captain to the Bosun, "so look for the bottle with the ship that sails the ocean!"