This thing, frequently cited in amateur circles, seems to not even be a rule at all, in any sense of the word. Go look at some actual photographs, commercial stuff that people have sold or "important art" or any category you like that counts as successful photography.
You will, most likely, find all kinds of stuff neatly centered. If you closely examine the 1/3 lines of the frame you will of course find something there. Is it the important something? Sometimes, more or less at random, yes. Usually not. Frequently it is a boundary between one thing and another, see below for more on this important case. Sometimes it's clearly just random stuff. Sometimes, to be sure, it is strongly the subject. More or less at random, the subject will fall pretty close to one or two of the 1/3 lines.
This horrible bit of anti-pedagogy should be stated more as the Not in the Center principle and should not be stated as a rule. Putting things in the center makes them look one way, and putting them not in the center makes them look another way. It is no more a rule than black-and-white is a rule. Both centering and black-and-white are effects which you can deploy, or not, as the case demands.
Photographers seem to think this rule of thirds comes from graphic design, so let's go look around and see if that's true. We run across amusing things like this advice: divide the page into thirds horizontally, vertically, or both. Now place important elements either within the zones defined by these lines, or in the case of a photograph, you might also place them on intersections of those lines. Let's think this over. So either within the zones.. or.. on borders between the zones. So. You're saying that I can place important elements anywhere at all, correct?
The painters, and later the graphic designers, created regions, and then placed things within the regions. Photographers are urged by the "pep up your snaps" people to place things on the boundaries between the regions, that is to say, in exactly the place where the painters and graphic designers teach not to.
There's something from graphic design here. Thirds are in fact a sort of pleasing way to chunk up a design, or a page. Graphic designs are not the same as photographs, and photographs are not, primarily, graphic designs (yes, there's some overlap, indulge me here). Photographs, in particular, have a subject and graphic designs do not.
The graphic design principle is in fact not to place things at intersections of the third lines at all, but to organize the page around thirds. The one-third lines are boundaries between things, in design.
Applying graphic design ideas to the problem of where to put the subject is silly, and it's simply wrong. We might as well try to apply ideas about depth of field to graphic design.