Pasta – Making pearl couscous in slow cooker

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.)

If you wish to continue down the road of a singular dish I would recommend the emulsifier. Incorporating the fat through blending it with some of the liquid in a side pan or pot with ground mustard would be easiest and most complete. You should be able to continue cooking without further separation.

However, if you are cooking the dish too hot you may be forcing some of the fat from the meat or otherwise loosing it from too much agitation and you will want to make any adjustments necessary.