There are certainly many cases where the rules apply almost absolutely, but there are almost always some exceptions, so the words "never" and "always" have to be carefully regarded. There are certain situations where you have to break a rule for the better good, just like if you have to break into someone's home if it's on fire to save their baby, or drive over the speed limit to outrun a racing tornado.
You first need to understand that much of the information out there is dumbed down. Those who write or teach realize that very few people will get a comprehensive education, so they make it as simple as they can with some rules to follow. Better safe than sorry. But most of the time, the finer points are missed or just disregarded. If you're one of those that just shrugs your shoulders, it's really best to find someone who knows what they're doing. Once they do what they need to do, then maybe you can do maintenance following their examples. If your tree hasn't been touched in years, or has been badly pruned, like most trees have, just once, hire an aesthetic pruner who knows what to do. Well worth it.
TYPES OF CUTS
Generally, you hear about only two types of pruning cuts, thinning and heading. Well, it's more complicated than that. Let's go on.....
THINNING is a type of cut where you remove a whole branch. Can be a huge branch or a tiny branch. It can be removed to a larger or parent branch, or to the trunk.
This is almost always the preferred cut. Confusion abounds, where people think that thinning means taking a lot of branches out of the middle of the tree and leaving almost all the branches except for the bare minimum major branches, at the very fringes of the tree. This is wrong. I'll explain why a bit later.
HEADING is generally the other cut you hear about. This is basically when you just cut off the end of a branch or tip of the trunk or main vertical leader. With anything bigger than a fraction of an inch, these cuts usually look terrible and are ill-advised. But once in a while, they're the best option or necessary. In most trees, if you make a heading cut, the end becomes blunt and stays that way, unless a new branch sprouts from that point. It may or may not depending on the situation. And if it does, it may go off in a weird direction that won't look good. HOWEVER, if you prune roses, pruning is generally made up of a bunch of heading cuts. In most trees, heading cuts are almost always avoidable if you follow a yearly proper pruning regimen and never let things get too far.
RE-LEADERING is a term you'll rarely see, and is generally reserved for those who know their stuff. Consider it a combination heading and thinning cut, from which you'll train a new leader or growing tip. This is done all the time in bonsai, but in landscape trees it has to be carefully considered. HOWEVER, if a branch is growing lengthy and lanky, and you want to promote tapering or side branching (ramification), cutting the tip back to a smaller branch or bud that's heading in the right direction would be smart. In reality re-leadering more like thinning, but back to a branch that either doesn't really exist yet or doesn't quite have enough oomph, even though you know it's there waiting to do its thing.
I use this technique a lot, since I'm very big on proportion and tapering, and re-leadering is frequently the only way to get there. Since apical dominance is so powerful, sometimes a branch wants to head straight out and never look back. This conflicts with creating important side branching and can weaken the branch and mess up the proportions. With re-leadering, you want to avoid cutting anything larger than about 1/2" in most cases, but you need to know what the results will be in any situation. If it's a huge tree, maybe a bit bigger, say up to an inch or more, and on small trees, maybe 1/4" is a bit too big. In any case, it's usually smart not to cut back to anything less than 1/3 the diameter of what you're removing. And it's also best to avoid taking off more than 25% of a given branch, since taking more than that can sometimes lead to die-off, since what's left doesn't have enough vigor, or it can sprout profusely since that removed energy has to go somewhere.
It's best to take out branches before they get to be no larger than about 2" in diameter, since above that thickness requires much more time to callous over. We're very careful not to use the word "heal", since unlike human skin that eventually becomes just like the original in most cases, think of a branch wound as an injury that remains either an open wound or a scab FOREVER. Trees are very susceptible to rotting within the branch column under certain conditions. The larger the wound, the more likely the rot, not to mention the ugliness. In all fairness, some wounds are actually pretty cool looking if they get a nifty ring around the collar. In bonsai, these wounds can enhance the apparent age, but these aren't always desirable. In most cases, it's advised to cut back to a branch that's at least 1/3 the diameter of the branch you remove.
TOPPING is when you simply cut off the top of a leader or trunk, with no regard to how it looks or will respond. This almost always looks terrible, and unless the tree is about to hit something, or has a bizarre protruding angle, or is dead or diseased, there's no reason to do it. EXCEPT if someone else has already done it, and in the course of making it right, you actually need to cut it back even more, as is the case with re-murdering Crape Myrtles that have been previously murdered. People do this a lot with Fruitless Mulberry trees too, and they always look terrible in the dormant season. No reason exists to do this. Why have a terrible tree instead of putting a good one in its place, and making sure it fits?
Re-leadering could technically be considered a form of topping if done on a trunk or vertical branch, though refined and with good reason. So anytime you see it stated that topping is NEVER ok, well, there are some exceptions. But generally it's a terrible thing to do to anything bigger than maybe 3/4" in diameter. And you'd better know what the heck you're doing.
LIONTAILING is the practice where people cut most branches out except for a few major ones, and leave almost all the branching at the fringes. This may actually look ok at a glance, but it eliminates most of the taper which is so important to branches, visually and for strength. What you really want in most cases is one heavy trunk, a few beefy major scaffold branches, a number of secondary branches coming off of those, and then finer and finer branches multiplying all the way out to the tips. There's probably some equation out there that makes sense of how many branches and what size they should be at any point within the tree, but it would be quite variable from tree to tree and always changing as a tree grows. I really find most people really don't get this, including lots of arborists and nursery growers. There's an artistic sensibility. There's also the need to understand how some branches are purely temporary. You may leave them near the base of a branch to increase bulk at that point, but remove them before the removal scar creates problems.
WATER SPROUTS are almost always to be avoided. These are the reactionary sprouts that shoot up vertically following pruning or injury, or other stresses, and some will happen with almost no apparent encouragement. In almost all cases, these are to be removed. I would sometimes leave these in place if I want to fatten up a branch at that point, and remove them before they're the size of a pencil. And if you get a bunch of these in the Spring, or after a flush of growth, it's a good idea to rub them off or cut them when they just emerge. Even if you rub them off, some will re-emege, and it may have to be done several times until the tree settles down. But in some cases, these sprouts may become a viable secondary branch. Probably not if they're coming right out of the top of the parent branch. But I guess a water sprout by definition is really a vertical shoot heading straight up. You'll see these in most vigorous trees, and perhaps the most obvious is plum trees and birches. I just pruned a mature Valley Oak that had some, in response to some major (not good) pruning by the utility company to keep a chunk of the tree from messing up phone or power lines. These companies lack much desire to prune correctly. They want to spend 1/2 hour removing a big chunk instead of 3 or 4 hours making it look good and equally avoiding the impending entanglement. My job is now to try to get that tree to balance out better over time. This will take a long time, since the tree is already at least 50 years old. But even a compromised Valley Oak is usually a pretty cool tree.
Well, maybe this strayed a bit from the main subject, but these are all connected points. Hopefully I made some sense out of it. I'm sure some professionals may disagree a bit on some of the terminology and remedies, but it's mostly dead-on, at least in practice.
I guess the most important thing to take away is, DON'T CUT ANYTHING UNLESS YOU HAVE A GOOD REASON AND A PRETTY GOOD IDEA OF THE REACTION.