Cold water shaving is a thing, many folks do it. Not me, but as they say... YMMV."Cold Shaving" is an actual thing. Ben Franklin among others promoted it. Cold water = stiff whiskers = cleaner cut. Cold water and skin don't promote as much bacteria growth as a warm environment. I kind of go with what ever water temp is available to me at the time.
I think the differentiation is cold water shaving and hot water shaving vs the proposed idea that just running a razor under cold water is magical.