The rule of thirds is a simplification of the golden ratio. If you take that box example (your first link) and you apply the numbers again you get a ratio of app. 60/40 (I love being an engineer: 61.8 / 38.2 can be 60/40 or 66/33 depending on what you need into be :) )....
I choose to read the underlying message as "don't sweat the details - composition is an art not a science".
We build up a subconscious library of images over the years that helps us to judge when something just "looks right". An image can work several ways, there is no definitive answer. I think that's why cropping (at least for me) involves some trial and error. Sometimes I resort to making virtual copies in lightroom so that I can try on different crops. Usually one jumps out before I ever get to comparing alternatives in the end.
It is the less common compositions that I often find more memorable. As Msmoto mentioned, creating deliberate tensions in the frame tends to grab attention: placing two object just that little bit too far apart, subject looking out of the frame or backwards instead of on the 1/3 line looking forwards / across the centre of the image, etc.
The standard 35mm frame and 1/3 : 2/3 or 60:40 proportions are very versatile but I enjoy a change from time to time. Wide panoramas, or something tall with a sequence of objects precisely down the centre line or along a diagonal, and once in a while you see a subject that's just made to go dead centre in a square frame. Not so popular these days as none of my digital cameras ever had a square crop option but 6cm square frames worked for medium format.
When an image makes it to print and has to fit a particular space in a particular building there's a a whole extra dimension of matching the print to the space and the selection of subject matter and composition in their turn.