First of all, ALL water damages hair because it causes the fiber to swell which in turn causes some cuticle lifting and cracking.
The rumor about cold water seems to be based on the idea that cold water somehow cause the cuticles to contract or tighten which makes hair smoother and shinier. Unfortunately, this is not true. There is some data, however, that suggests that cooler water temperatures can reduce irritation of some surfactants on your skin.