I guess "The Cold War" is very specific to a historical period, and the term "grey zone conflict" is a generalization of what went on there. Also the Cold War involved lots of proxy wars, not thing "grey zone conflict" necessarily does.

Also, I'm starting to feel like the vagueness of "are we at war or not?" is an intentional feature that gives people in power leverage to gaslight the public. That applies to both cold war and grey zone conflict.