Let's say there's a bully, and he just takes and steal stuff. And then there's a big guy seeing that and telling the bully to fuck off and let the kids play without fear.

Both are power moves. Do you think they're the same thing though?

Yes, power is good way to bootstrap cooperation based on trust. This is a much more effective form of cooperation as you waste less energy on paranoia and watching over your back. You can also base cooperation on submission but then you have to constantly be at an all out war to protect your regime. Much less efficient or reasonable.