The color is more of a weather indicator than a dormancy indicator.
It's a response to cold temperatures; the plant makes these pigments divert the solar radiation away from its chloroplasts into molecules (like anthocyans) that can take a good shake-up. See it as a pair of molecular sunglasses.
Due to low temperatures, the biological functions are down and letting solar rays hit the relatively expensive chloroplasts would cause them to overload and break (oxidize). That would result in total foliar death or bleaching. So to protect themselves, they make these colorful pigments and push them near the foliar surface.
Pretty cool when you think about it! Because human skin - in principle - does somewhat the same by tanning.
Plants can still grow and be biologically active when these pigments are present, and they stick around for a bit when temperature goes up again. I'm saying this because I've seen people being fooled by it; their plant was out of dormancy for a month but the pigments were still there because of the cold nights.
In younger plants, we see these pigments all over the place, which is why they can range in all different colors in the red spectrum. I haven't looked into the science of it for over a decade, but plants might have picked the red spectrum because that's where infrared is the closest to, the molecules might serve a double function to both divert harmful light and capture infrared to generate some warmth. The part about infrared is speculation on my end.