I would really like to go to bed, but the slow cooker isn’t done

It's really no different from cooking something on a stove: if you have no liquid at all, things are probably going to start getting too hot, sticking on the bottom, and maybe burning. As long as there's some liquid, you'll be fine. The important thing is to add enough; if you keep having to open it to add more, it won't stay hot. This is basically the minimum level: enough that it won't boil dry in an hour or so. If you're cooking something really simple like a piece of meat, using too much water just provides more water for the flavor to get diluted into.

This is precisely what the first page you linked to says: if it's not full enough, it might all boil off. The writer is just way off about quantities. I suspect his cooker's lid doesn't fit well, or he's unnecessarily cooking on high. I don't think my slow cooker would boil dry from half full in 8 hours on low, and there's no reason to put it on high if I'm cooking that long.

The recipe at the second link isn't exactly empty - a can of cranberry sauce will provide a good amount of liquid. It also might not be a great recipe. On that note, I might suggest you look for more authoritative recipes. Perhaps check out a slow cooker cookbook from your library (these tend to be pretty common) to get an idea of the kinds of things you can do. (Of course, if all you want is a plain piece of meat, I guess you don't need much.)

It depends on the particulars of the recipe and the ingredients and the slow cooker itself.

You say you are using chicken or beef, but are the pieces of meat small or large? Is the meat lean or fatty? I've had problems with overcooking large pieces of lean meat, such as chicken breast, but smaller pieces and/or fattier pieces seem to hold up better to extra-long cooking.

Another issue is proportion of liquid. For example, soups tend to work better in your scenario than "drier" recipes.

And, of course, the slow cooker itself can make a difference. I assume you are talking about 9-10 hours on low, not on high. But low itself is not likely to be entirely consistent between cookers. For example, a large slow cooker is likely to put out more total energy than a small slow cooker, so for extra-long cooking times, you are probably better off using a small cooker with a full pot that a large cooker with a half-full pot.