Before discussing what type of breathing generates power, you have to discuss how the body generates power. And even more importantly, how martial artists apply the principles of power.
Martial arts is less interested in the physics definition of power--mainly because it is of little practical use. Typically, a martial artist is interested in the force transferred to a target. Crushing a coconut requires roughly 1600 psi (pounds per square inch), which is 100 psi more than it takes to crush a skull. The martial artist isn't interested in the physics that cause the kick or the punch to generate the transfer of force, although that is done with power.
In the realm of physics:
- Power is force over time.
- Moving a 400lb object 2 feet in 5 seconds requires less power than moving a 400lb object 5 feet in 0.5 seconds
In the realm of martial arts:
- Power is how much force is transferred to the target
- Knocking a 300lb man on his butt requires more power than knocking an 80lb child
How the Body Transfers Power
This is the short summary as applies to martial arts. You can have two martial artists that can move their hands and feet with the same amount of power as defined by physicists, but the impact on the target is different. There's a few reasons to explain this, but if we eliminate external factors and focus on the budoka (martial arts students) the difference is the stability of the budoka's base. The target has force, even if it is stationary. The target absorbs some of the force, but due to surface tension, and how stable that object is, some of the force is pushed back. The budoka with the most stable connection with the ground is going to be in a better position to overcome that push back.
There is a concept that power flows from the earth through you into the target. It sounds mystical, but also has application in a physical description. When a budoka is in a solid stance, any force applied to him will be transferred to the earth. The earth has a greater surface tension than any of us, so assuming the body is rigid, the earth in essence pushes back.
The same applies when transferring the physical power of a punch or kick to a target. The more rigid the rest of the body is, the less the push back from the target is going to overcome the force applied for all the same reasons that a good stance helps the budoka. If any of the muscles used in the technique are not tense, then it absorbs some of the push back and diminishes the force applied to the target.
How this Relates to Breathing
There is a reason why we are instructed to kiya when we strike. In order to form the kiya, our core needs to tighten quickly and force an exhale. When our core tightens, it is more rigid and it becomes a strong component in our stance. This applies to both receiving and delivering a punch. Even if you don't make an audible noise, tightening the core at the moment of impact provides a good balance between stability and freely getting the breath you need.
Different arts have slightly different emphasis, with different focus on breathing and how it applies in the spirit of the art. I'll only provide a couple of examples that I'm aware of, as I wouldn't know where to find out all the answers.
- Tae kwan do, and similar arts focus on one-hit-one-kill. Even a block is a strike. To that end, the kiya happens with the strikes and the blocks.
- Go Ju Ryu, and some similar arts focus on prolonging your energy to fight. Breathing becomes an important part of practice. It's blocks are more of a parry to deliver a counter attack. On the block, the budoka breaths in, and on the counter strike the budoka kiyas.
- Jujitsu, and some similar arts focus more on grappling than strikes. The budoka needs to learn how to breath while they are under pressure, as well as how to apply the techniques.
Just to throw something else in the mix, several martial arts teach the student how to fall. Typically, when the student hits the ground then need to exhale. That exhale releases tension from the body and helps prevent injury. It is also a normal reaction to the impact of hitting the earth, so it is best to learn how to control it so that when the body hits it is not too relaxed.
Regarding where the breath comes from, and all the different methods you referenced, that is highly dependent on which martial art you are studying. There are different philosophies regarding combat, and how best to survive. Applying the breathing techniques from tae kwan do in go ju ryu will leave you in a disadvantage if the fight goes longer than you expect.
With that disclaimer, breathing through the chest (chest rises and falls when you breath) is the least ideal way to breath when we need to control force. The same thing applies with singing. If your breath comes from your gut (same as diaphragm but more descriptive of how the body feels when doing it) you will usually do much better.