The problem with telling folks to brace or push out is that its not entirely clear. First there is bracing which is tightening something one would do if one was expecting to be punched in the stomach. 2. Then there is drawing in of the abs, Stuart McGill doesn't advise this, he says brace the abs. So far so good, brace your abs by tightening them expecting to be punched in the stomach.
Now, powerlifters tell us to push the abs out by expanding the diaphragm they do this against a belt worn around their waist. The whole idea of the belt is to create intra abdominal pressure. Pushing out or bracing the abs while different, I think, do the same thing , that is to say they create intra abdominal pressure.
To add another aspect to this is taking a gulp of air, and taking a gulp of air and closing the glottis. Similar actions. The closing the glottis , i believe, is the karvonian movement. Taking a gulp of air and holding is might be the same as the latter or karvonian movement but with less emphasis. I prefer to do the taking in of air and holding it not pushing against the glottis since pushing against the glottis( holding your breath and trying to expand your diaphragm creates more pressure. One feels as if one could pass out.
Also if you take in a gulp of air and hold it one can still breathe through ones nose. That's my take on this matter. Perhaps i need clarification by one who knows.