Active shoulder is basically just moving the shoulder in the direction opposite that the force is trying to move it. So when holding something overhead, the force is pushing down, so an active shoulder is resisting that by pushing up. When hanging from a bar, however, the force is pulling the shoulders up (or the body down, as it were), so an active shoulder would be pulling the shoulders down.
The overhead active shoulder is more of a concern in my opinion--more of an injury risk. It reduces the pressure at the AC joint and makes the shoulder stronger and more stable. When hanging, the only reason you'd really need an active shoulder is when initiating a pull (as in that is the first movement that should occur when starting to move up)--if you're just hanging, particularly to stretch, allowing the shoulders to relax and elevate relative to the body is fine if not good.