I recall from some birdwatching stuff I've read that the number of "dees" indicates threat levels but only of certain predators. They have to feel safe enough to make noise. Acipiters (cooper's hawks, goshawks, etc.) can illicit a quick hush by all. Depending on your woods, the threat may sometimes just be a rival male or a corvid getting too close to their nest. It's a good indicator of mood but not of the specific threat. Sit in the same spot long enough and you may learn what your particular chickadees will do for different threats. Deer have the advantage of getting to do that 24/7/365. I'm sure they have a working knowledge of the subject.
Birds are often the eyes and ears of the forest. Just about all animals are mindful of their mood. Bird brains, while small, have incredibly fast processing speeds -- likely required for their flying abilities. They can observe, assess & react to a potential threat much faster than anything else in the woods.
Seems plausible that some deer and, by the sound of it, YOUR deer are very well-attuned to the chickadees. In the coastal woods where I sometimes hunt, the local blacktails are attuned to the invasive european collared doves during their roosting times. If the doves flush before dawn, I won't see a deer all day. If I wait to hear the doves fly before leaving camp, I usually see a few.
I suspect that although hunter orange may keep us hidden from deer, the birds are keenly alert to it. I've been the unhappy recipient of some light mobbing from juncos when wearing hunter orange that I just don't ever get when I'm not wearing it. But that may also just be because I only wear the orange in the rifle season. But that's just speculation.
It's fun to learn & think about these things, though. As the old saying goes, "The ocean just gets deeper the further down you go."