Relevant xkcd.

We have a family of bald eagles that live in my area, and every winter they lay an egg or two, then sit on it till spring out in the elements. They seem to not really care one way or another, even in the snow, but I'm not a birdologist.

our rescued ducks like the rain but the chickens will get soggy. ducks are pretty much waterproof

I have birds that I keep outside in the summer, sometimes they will just sit in the rain even though they have plenty of covered areas. All birds have an oil gland that they use to run an oil over their feathers to keep them in good order and different levels of waterproof depending on species. Most birds can sit in the rain for a while without actually getting wet.

But ya, birds can get wet and cold but they are usually adapted to their environment to not get too cold or too wet.

Anecdote time!

Our rooster refuses to stay dry when it rains, unless it is literally freezing.

He's a big ol' motherclucker, and the closest he gets is jumping on the grill, which is half under a tree so that big drops don't hit him direct when it's coming down hard.

If it's a regular rain, he just goes about his business after about fifteen minutes of what seems like bathing, where he fluffs up and flaps his wings while stomping and then preens for a bit. Then, he just hunts for bugs and happily clucks when he finds them. Sounds like a popcorn machine.

Probably depends on the bird.

Many birds live all year round in cold, rainy environments. Their bodies developed to be able to handle it, so either they are kept warm by their feathers or they shelter in warm places.

Other birds, such as toucans, are native to tropical environments and would probably be very unhappy in the cold.

I have a balcony with bushes, and hummingbirds always perch on the inner twigs to get out of the rain, eschewing their usual dogfights around the feeder.

In my anecdotal experience, some birds stay out of the rain and others don't seem to care. Crows don't seem to care, but the finches (or whatever the smaller birds super common around here are) tend to take shelter.

my macaws hate it

Can't speak for the cold but many birds like water because it keeps them clean/parasite free, and that's why they bathe in bird baths and puddles.