Why usually is the word “halfway” used with “down” rather than “up”?
I doubted that. I can't imagine that there is some kind of rule to use down rather than up. It's possible, but I didn't see one. I looked to Ngram.
So the usage looks roughly even, at least in recent years.
One reason you might use one over the other is perspective. For example, in a larger context (like a story), if you want your audience to visualize themselves looking down at the nest (fixed, not moving), then you might use halfway down. If you want your audience to visualize themselves looking up (at the base of the cliff) at the nest, then you might use halfway up.
In a slightly different use, imagine a group that is at the top of a cliff and they have a friend climbing up. He reaches the halfway point and one of the friends reports "He's halfway up the cliff." In this case, halfway up implies that he is moving and he is moving up. You can construct a similar case for halfway down.
The given exercise/example is so brief that I don't really see a difference. It's possible that the author was "looking down" or wanted the reader to look down, but I think either choice, down or up, is equally likely and acceptable.