I've been doing a few things lately. Reading nineteenth century books on composition, reading web pages on composition, and doing some research in Google Books to see when various things turned up.
Until about 1940 there is no royal road to composition. Teachings about compositions are about lines and opposing lines; about balance of masses and colors; about unity and variety. Many basic principles discussed, effects of one thing and another are laid out. Many many examples are uses to illustrate these ideas, usually examples drawn from contemporary and classic "good" painting that meet with the author's tastes. In the end, though, it is clear that no formula is to be given and that the artist's taste is ultimately what rules. Principles can be memorized, but applying them requires taste, and taste is learned by looking at a lot of pictures.
Things like Golden Ratios do seem to have turned up now and then over the last couple thousand years. While it's rarely clear in any individual case that the Golden Ratio or Rectangle was in use, it is certainly credible that it would have been used now and then. It is a ratio and figure that has been known and mentioned for quite some time (2400 years or so) and it's the sort of thing numerologically inclined artists might credibly have picked up. Not every artist, or even most artists, are so inclined. There are surely spans 100s of years long containing none of people doing significant art. To say that most art, or even very much art, embodies this thing is simply false. The examples trotted out are, tellingly, always the same: a couple da Vinci's, the Parthenon, and Mondrian. The Mondrian is particularly hilarious, since many of his paintings are made up entirely of a huge number of rectangles of various shapes. Of course approximations of the Golden Rectangle will turn up. Extensive rebuttals of all these claims have been written by people better qualified than I.
Regardless, I find it credible that Golden Rectangles and Ratios appear in art, here and there, so I let that stand.
In 1940 there is a little book tellingly titled A new approach to pictorial composition aimed at photographers. This is the first reference I can find to this horrid business of placing the subject at the intersection of one-third lines. This dumb idea vanishes again until the 1960s, finally appearing in the Feb 1970 issue of Popular Mechanics. (Popular Mechanics? Really, need more actually be said at this point?) At this point it seems to more or less take off and usage of the phrase and teaching of the method ramps up steadily.
The rule of thirds is, contrary to what we are routinely told, not an old idea. Painters have not used this thing through the ages. It was invented in or around 1940, for photographers, and is still mainly used by photographers.
There is an eighteenth century notion called the rule of thirds as well, but it is about allowing one region of a picture dominate. Two-thirds land, and one-third sky, or vice versa. It is completely different from the thing photographers use, as it says nothing about placement of subjects, and in fact does not use a grid at all. It's entirely about proportion and balance, not about "put the subject here."
To be clear: The "rule of thirds" as introduced to photography by Popular Mechanics in 1970 is not the classical rules of thirds and has nothing in common with it except that the ratio 1:3 appears in it.
The Golden Triangle, another commonly cited rule of composition, is a complete mess. There are at least three differing definitions for this particular pig. There is a mathematical figure with this name, an isosceles triangle where each of the two equal sides is in the golden ratio with the (shorter) base, and a second mathematical figure with a different definition. I suspect this second one is simply a mis-remembering of the first one, which has been repeated here and there by clumsy dolts. These seem to have little or nothing to do with composition although many confused would-be authorities on composition cite them in a muddled sort of way. The compositional rule is this business of drawing a diagonal, and then dropping a perpendicular to either of the other corners of the frame. The intersection of these lines is where you stick the subject. This thing doesn't seem to appear until the 1990s or later, and always in books with titles like "Better Composition With Your DSLR".
The Golden Spiral follows a similar pattern of usage, but if anything starts out later, really coming to prominence in these dumb books around 2006. The Golden Spiral as a geometrical figure is a few hundred years old, and approximations to it appear in nature, apparently. Its use as a compositional device, though, is fully modern and again primarily aimed at photographers.
A word on web pages with advice on composition. These things are universally terrible. Most of them contain wild technical errors, simple errors in geometry. They draw lines on a rectangle, and make claims about proportions that are not mathematically possible. They draw one-third lines in places that are not at one-third. They draw random rectangles on things and then cheerfully proclaim "look, so and so clearly used the Golden Rectangle here!" and so on.
It's clear that all this business is about providing simple and easily mechanized rules for composition to technically inclined nerds who have bought a camera. Painters are not particularly interested in these things, since they're already on board with the idea that art requires a bit of taste and thought, that you cannot reduce it to a technical procedure. Painters still refer to the ideas found in those nineteenth century texts, updated and so forth to reflect the current fads in art. Painters are also presumably aware of the Golden Ratio, and may even use it from time to time. Mainly, though, they use ideas of visual balance, of unity and variety, and so on.
In virtually all cases these little drawings and maxims boil down to this: do not stick the subject in the center, or at the edges. This is disposed of in real books about actual composition with a phrase like "the center of the frame is the weakest point" and then we move on to actual composition.
In short, when someone tells you that these rules of composition are old, they're simply wrong.
All of the rules which advise you on where to stick the subject within the frame are fully modern, and were devised by photographers for photographers.