First, let me say that I pretty much have never gone along with the crowd. I was born in the early 50s, so I saw the craziness of the 60s and the ridiculous fashions of the 70s. I never engaged in that stuff, so perhaps that is why I have never understood the emphasis on the rule of thirds, and I never will. I always have to come to my own conclusions, and don't like being told how to do something as if I cannot think for myself. In fact I feel that the rule of thirds was created by someone who felt they were a bit smarter than the rest of the crowd, and this was their way to educate you quickly. It's origins date back decades, but who is to say that it is correct? In my opinion it is not. The rule might be good for many of us on the right occasion, but it assumes that everyone takes pictures for the same reason (one picture is no different than the next), and/or has no idea how to compose a shot. I take my picture based on a lot of things, and while taking the picture I ask myself some simple questions. First, and foremost is the question why am I taking the picture? Is it a picture of my wife, or a picture of the ocean with my wife in the picture? That makes all of the difference in the world. Sometimes you take a picture to put yourself or friends at a location (like vacation pictures). Now I am not saying that many of your pictures won't be using the rule of thirds, what I am saying is that you should think about what you want to show people when they see the picture you have taken. If you try to apply the rule of thirds all the time you are going to have some crap pictures. The same rule is used in video and I actually saw a video demonstrating this with a boring background. The talent centered version was actually better because in the rule of thirds version you could see more of the boring and busy background. The guy making the video didn't see that. Pretty funny. I have experimented with this a bit and the average person sees little difference between a centered subject and rule of thirds subject. In those cases where the background is interesting they sometimes prefer the thirds version, but not always! This is especially true with picture of people you know. Some people tend to look at the person and pay less attention to the background (and vice versa). Sometimes my wife doesn't like a picture because she doesn't like the outfit she is wearing and pays no attention to composition.
Let's take a look at the picture of the squirrel. I was out for a walk to get some fall color pictures. I love the curiosity of squirrels and some of the crazy stuff they do, and I never fail to take a picture of one when I have the opportunity. In this case I wanted to get the picture of the squirrel with the fall colors. To do that I had to place him to the left of the picture. I didn't place him to the right because the background wasn't as good. So in this case the picture was driven by the background and color. If I centered the tree I felt it was too drastic, and almost created a split in the picture. The anchor point to the picture is the tree and the subject is a squirrel on a fall day. I got the photograph I felt was composed the best and there it sat in my picture folder for 5 years before I did anything with it.