in MA you actually do the same. You do the same in a lot of lifting as well, many people breathe in and hold it until a certain point in the lift when they feel they can breathe out. For example for me, I start breathing out when the bar gets over my knees in the deadlift, because that's when I'm over the tough part. and it's not breathing out that makes it powerful, it's what you do before you breathe out. To create a ****load of sound, like KIAY, when you punch, you have to press a lot of air against a closed glottis and then release it. Pressing air against a closed glottis is the valsava maneuver and it makes you stronger. I think that's where the yelling comes from. You probably let it go into a kiah because breathing out relaxes you, makes your muscles and joints able to reach a better ROM, which you want when kicking or punching someone. but yeah, I'm pretty sure they do the valsava maneuver in MA as they initiate the punch (IE, rotate) then breathe out when the power has been gathered. to put it like that. Also, it hurts less to take a blow to the stomach when there's no air in there, so you're safer from counter attacks with breathing out.

I personally follow the rule that you exhale on the exertion of a particular exercise. This has served me well over the last few years.

This might be TMI, but I actually got hemorrhoids this summer, never having had them before, and I am fairly confident it was from lifting weights. It happened twice when I first started squatting. Haven't had any trouble since. Not sure if it was related to my breathing.

Just adding my own 2 penny worth. I believe its only exercises like squats and deadlifts where you hold your breath during exertion due to maintaining core strength to protect your lower back. I would also agree that in general controlled breathing out (as not just sighing) on the effort part of the exercise does seem to increase power. I certainly wouldn't want to hold my breath doing heavy benchpress. That is going to lead to burst blood vessels. You get red enough from bench press already! Video yourself, its not pretty!
You don't need to be pushing 700lbs to get a blown eye. You only need to be pushing what is really hard for you. My friend did it with about 280lbs. I really do feel that holding your breath during bench press is a bad idea. More so than perhaps any other exercise due to its nature.

I have nothing to add about the piles discussion than my sympathy......

ive alwasy done #2 thats how i was taught so idk

I use method #1 personally can cannot picture myself doing really heavy weight for me while doing method #2. Method #2 is great for light weight or warmups but near impossible while lifting a 1 max rep.

I use method #1 but modified slightly; when the bar is on the way back up, I do breahte out gently until the bar is at the top and take a deep inhale for the descend and 1/4 of the way back up (repeat).