Asymmetrical Balance vs. The Rule of Thirds

As I was looking through our book, I came across a section titled "Asymmetrical Balance" (page 79). This section talks about how it is important to keep balance in a picture by having different sized objects or points of interest on either side of the photo you are taking. This all makes sense, but then I thought of something we learned in Doug's class (FA 332): the rule of thirds. This rule basically is a guideline for keeping objects out of the middle of a picture and keeping them closer to one of the corners of the picture, shown by this diagram:

My main question is how do the these two work together? Would it be better to have your focal object alone in a picture or have something on the other end to give it balance? Here are some more examples:

This picture obeys the rule of thirds, but not asymmetrical balance.

This one, however, has asymmetrical balance and has the rule of thirds, somewhat. I feel as if the pawns on the right kind of ruin the rule of thirds seeing as some of them are basically in the middle of the picture.

So, should you always follow the rule of thirds? Or should you always follow asymmetrical balance? Or should you try and always use both in a picture? I have no idea. This whole photography thing is still pretty new to me!