I don't know much about control, but apparently Emanuel Todorov knows about the basics of Boston Dynamics control in this video https://www.youtube.com/watch?v=7enj1FGoYwg around 14 minutes. The paper is this one https://repository.upenn.edu/ese_papers/686/. I don't understand most of it, but it may help if you have some knowledge. But from the few things I understood, not much is programmed.

Also, I'm watching some lectures for a MIT underactuated robotics class (https://www.youtube.com/watch?v=VeEqtTgDXFc). The professor says in one of the videos that the people who end up creating Boston Dynamics also took that course. From what I watched until now, nothing is explicitly programmed (by some vague definition of "explicitly programmed"), you just find some inputs that makes the dynamics of the model behave the way you want.

Almost certainly the birdlike movement is the outcome of the robot dynamics, and not some hardcoded rules. But things like avoiding hitting obstacles or estimating the distance, I think that needs to be explicitly programmed (and then used as input to the model such as "move the end effector to that position").