When is it OK to crowd the pan?

First, you don't crowd the pan when sauteing. If you do it, you are no longer sauteing, because your food doesn't come in contact with the surface frequently enough. You also can't really keep the separate pieces hopping, the best you could do would be to throw it around as a mass (like a pancake turning), which means they'll touch the pan with one side only, if at all.

Second, you don't crowd the pan while shallow frying and pan grilling or working on a griddle. If a piece of the food is not in contact with the bottom, it doesn't fry at all.

Third, you don't crowd the pan when deep frying. If you do, the food cools the oil down and it starts seeping into the food, because it's not hot enough to crisp the surface within the first few seconds.

You can crowd the pan when you are sweating the food. In fact, if you use too little food, it will probably start pan frying or at least roasting, and won't be able to sweat.

You can also crowd the pan when you are braising the food. Else you would have too much water and will simmer it instead.

Bottom line, crowding or not crowding is part of the technique. You decide when to use it based on the technique you want to use. If you don't do it right, the technique doesn't work.

I agree with everything rumtscho said in her answer.

I would add the "crowding" may be a little vague because the density of food in the pan may vary significantly depending on what the ingredients are.

It's also worth noting that definitions of things like sauteing and sweating can vary a bit from person to person, and there are cooking techniques that seem to fall "in the middle" between these or can't really be classified under strict definitions of one or the other. They may be less effective in producing specific results, but they are often more practical for those who want to cook reasonably quickly without many stages for a single recipe.

For example, take your middle recipe. The quantities of chicken, onions, and mushrooms added will not make this a "strict" saute, which should generally allow all food to be in contact with the pan so conduction is the primary heating method. On the other hand, it's not really a "strict" sweat either, which is usually done with lower heat (not "medium high" as in the recipe) and isn't generally done for browning (unlike the recipe, which states to "brown the chicken well"). Some might call it pan-frying, given the bacon grease and large pieces of chicken, but the veggies will make this mixture more moist than typical frying in the first few minutes.

So what is this recipe doing? I'm sure different people would have different opinions, but it's a very common technique too. Often one throws a lot of ingredients in a pan that initially release moisture ("sweating"?) over high heat, but after that initial burst of moisture evaporates off, the ingredients will begin to brown ("sauteing"?), though they might require periodic stirring to keep changing the food in contact with the very hot pan surface.

To try to answer your question -- if you want Maillard reactions and browning, you either need to get the temperature of the food surface significantly above boiling or potentially cook for a much longer time. What amount of heat and the amount of acceptable "crowding" will depend on how quickly the food releases moisture and how long it takes for any released moisture to evaporate off.

For example, if you're cooking hamburgers in a skillet, they will often release a lot of moisture quickly. To avoid "crowding" and get acceptable browning you not only want all of the burgers in contact with the pan surface, but you also want to have significant gaps between the burgers to allow any water that might be released and pool around the burgers to evaporate as quickly as possible. Density of food pieces here is quite low with large gaps on the pan surface.

But in other cases you might have much smaller gaps and even food piled up a few layers thick and still achieve browning. For another example, if you're trying to create caramelized onions, you often don't want to just brown the surface of the food (as in burgers), but also cook them thoroughly for a long time to get adequate Maillard browning throughout. In that case, you might not want to saute in a strict sense where all food is in constant pan contact, since that might burn the onions prematurely. So, you can pile up the onions a bit more in the pan without it being "overcrowded," since after the first 10 minutes or so of cooking, the liquid release starts to slow down significantly and the temperature of the food will rise. Periodic stirring will allow enough pan contact to encourage browning.

Bottom line: In most cases, if you want browning, you need to avoid crowding so much that liquid pools in the pan. Then you're really steaming or sweating or even simmering/boiling (if enough liquid appears). If the liquid is evaporating fast enough that the food in contact with the surface can get above boiling, it's easier to get browning. While it's often more effective to do a "strict" saute or sweat or fry or whatever for very specific effects, recipes often combine such steps together to make the cooking easier -- for example, as in the recipe above, starting with a "crowded" pan that sweats and softens the food, but then cooking over high heat long enough that the food will eventually dry out and/or be in prolonged pan contact and allow for some browning. Other recipes might deliberately use an "overcrowded" pan with food that doesn't release a lot of moisture but piled high over high heat (and stirred frequently, like stir-frying), which effectively combines some of the sweating and sauteing effects and may allow you to get your cooking done more quickly and efficiently (i.e., without various stages and multiple pans).