Either or both. It really doesn’t matter. We do it all kinds of ways for all kinds of reasons. Whichever is easier for you works tremendously well.
Doesn't matter to the birds. It just depends on your set up. If you have an area in the run to keep the feed dry, outside is fine. If the birds are going to be confined to the coop much of the time, inside is best. Personally, I have my water out in the run, and the feeder in the coop. I have a lot of rain in the winter and can't keep the feed dry, so inside it is. Too much of a pain to try to fill the waterer inside the coop, I had chickens escaping through the door while the hose was there. So, water outside where I can fill it through the fence. Whatever works for you!
I keep the water outside as long as possible! It will spill and if i don't clean it quick enough or realize it spilled, mold will grow.