I've been over this a few times here and there, including on TPF, but just to re-iterate:
There are at least two distinct rules of thirds, a distinction wikipedia was missing last time I checked, but it's important.
Both versions start out the same: Divide the frame into thirds, vertically, horizontally, or both.
VERSION 1: Now places things into the regions formed by the lines at 1/3 boundaries. That is not on the lines.
VERSION 2: Now places things on the lines at the 1/3 boundaries, ideally at an intersection of two of them. That is not in the regions
The two versions give exactly contradictory advice on where to place "the subject" and other important objects and masses. The two versions are the precise opposite of one another.
The first version existed at least as early as 1797, it appears regularly from that point onwards, and still appears in contemporary textbooks on design and painting. A quick perusal of.. any visual art collection whatsoever.. will show you that this general design of dividing into thirds and organizing masses within the grid so formed is extremely common in whatever we consider to be good art, as well as good design. The New Yorker, for crying out loud, follows this design. See also Mona Lisa, Migrant Mother, The Storm on the Sea of Galilee. It takes literally seconds of work with google to find example after example.
The second version appears, as near as I can determine, around 1940, and nobody except photographers pays it the slightest attention. Painters occasionally mention it, in order to openly mock it. While there is certainly "good art" that follows the general design, and has something or other of important placed more or less on a 1/3 line or the intersection of two of them, it's nothing like the quantity of art that follows the first version. In fact, good pieces which follow this second rule with any degree of precision are surprisingly uncommon. Try googling the names of various famous painters and checking the pictures that turn up. You'll probably find something eventually, but it'll take a while. Camera club output, locally produced postcards and calendars, on the other hand...