Fox can grab chickens and leave no signs. I've seen it multiple times here. They seem to get a good grip the first time, and they leave without the bird struggling. Coyote on the other hand like to stop and have a taste which makes it messier.
I personally have never had coyote attack my goats. They are all around us. I know a lady who did have one of her goats killed by coyote. So I think it all depends. They seem to prefer easier prey.
Both fox and coyote have taken chickens, and female muscovy ducks. They don't bother the adult...
Fox can take chickens without a trace. Coyote always leave a mess of feathers. I too have lost a lot of birds to fox in the past 2 years. Not much you can do but keep your birds locked up unless you are around. Fox won't mess with goats. Coyote may.