Well you're talking about the specifics of when and how to breathe, but maybe you really should be asking about why one breathes and what are you trying to do with it.
Generally speaking, when one exhales, this creates tension in the abdominal area. At the same time that your abdomen is tensing, you will also create tension in the entire core (the abdomen, the obliques, the lower back, the lats, etc.). A tense, strong core holds the back in place, which in turn connects the upper body with the lower body and allows them work together so that you can put your entire body into your punch, block, or whatever other technique you're doing.
This doesn't come naturally. This is something you're training to do. Over time, it will feel like it's natural.
Breathing out serves another purpose as well. It allows you to take hits to your solar plexus. If you've ever been hit there, you know that it's serious stuff. If you get hit there right as you're inhaling, it can cause you to stop breathing altogether, fall down, and go into the fetal position, helpless and in pain. It's what we know as "getting the wind knocked out of you".
So you learn to exhale whenever you're in a particularly risky situation. Anytime you're performing a strike or a block, you're generally more vulnerable to strikes to your abdomen. Why? Because 1) you're closer to your opponent now (he can reach you), and 2) because your hand is now occupied instead of being free to protect your abdomen.
Another time you exhale is when you're performing a breakfall. There are rolling breakfalls and there are slapping breakfalls. When you hit the ground, even if you're attempting a rolling breakfall, you risk hitting so hard that it feels like a hit to the abdomen.
Having some tension in the core along with breathing out just as you're impacting will prevent getting your wind knocked out during a breakfall. You can have your wind knocked out even if you land on your back, by the way, because the impact sends a shock wave through your entire body. Tightening your core muscles can also prevent damage to your bones and especially your neck and spine in most breakfalls. In some breakfalls, however, you'll need to loosen parts of your core, but that's a different subject.
As far as "holding back" some air while you exhale rather than exhaling all the way, that's generally a good idea. You often don't need to exhale completely to gain all or most of the protective effect. And your body needs that precious oxygen during a fight.
So you can experiment a bit to see what happens when you exhale all the way vs. exhaling part of the way. Find the point at which you're most comfortable. You just need someone to slap or punch your solar plexus for you (you tell him how hard to hit of course). It's unpleasant, but enlightening.
Another thing to factor into your overall fighting game plan is that you can't just take quick inhales and long exhales. Doing that will drain your oxygen levels, and that will cause you to get tired quickly. So you try to breathe in and out slowly and continuously, and only exhale in short, strong bursts when you need to.
This is harder than it sounds, because fighting causes you to panic a little, and your adrenaline will surge, causing your heart rate to increase and your breathing to get choppy. You need to get control over your emotions and calm down a bit first. Then you'll have more success.
Hope that helps.