Thank you all for your responses!
What I'm getting out of this is:
Work the muscles in the abdomen, but don't get tense.
It's about connecting, training the body, and also training ones awareness. It's a beginner step to having a connected body. (whatever "connected" means).
New questions: How important is posture? Is it ok to work on this when lying down?
connectedness: notice in the video you linked how he always keeps the body slightly extended and full, but not tense. If you do that with your arms and then add the breathing, you will feel that same stretch in your torso that you should feel in the arms and they kind of come together to connect your upper body as one unit. With enough of that combined with the various visualizations/intent you will get to a point that you feel that same tingly stretchy feel with having to stretch and/or breathe.
lying down: I wouldn't do it that way. One of the benefits I've found in doing reverse breathing is using it to help identify my center. When I breathe now I feel the expansion/pressure in my pelvis and I think laying down might be counter productive to that, initially at least.
Posture is important, but I don't think there's anything specific to breathing that's going to help you identify good vs. bad posture. I used other things to help gain some semblance of a relaxed but stable posture. Once I felt that and could comfortably get there without a lot of prep work, I know what to shoot for when doing anything else, breathing or otherwise, but I don't think I would have found that same stability from breathing alone. YMMV on that though.