Another good post Bill. When I studied TKD I don't remember being taught any breathing other than when winded, breathing in while move one foot forward, and raising our arms sideways, then reversing for several repetitions until we had our breathing more normal. Sounded silly when I was first told to do it. I quickly found out that I was silly as it worked very well.
In the Hapkido I studied, part of our warmup preparations was meditation and breathing. With our hands at our hips, we breathed out as we moved our hands straight out, and breathed in as we brought them back. Then the same breathing but the hands were moved up, then out from the shoulders and in front of us with our arms straight, then hands were moved down slightly to our inner thighs, same breathing. All the time concentrating on our Tan Jon (the area about three inches below our belly button. I don't know what it did for anyone else, but as time went by it helped me gain gi.
Breathing out when you are about be struck, especially in the stomach/solar plexus area, with a loud kiai does aid in resisting the blow, and as importantly, or rather as part of it all, not having your wind knocked out of you. I was also taught to kiai on landing from a throw, again to resist any damage.
As you say, breathing is indeed one of the most important things we can learn in martial arts. Thanks again for teaching all of us again. Reverse breathing looks interesting and I need to practice it to see what I can learn from it.
We call it the tanden or dan tien. Same location. Some take it further than this set of basics and begin to speak of qi or chi breathing, but that is more than I intended here. And in that area, I have so much to learn, I cannot really speak of it.