This is a wonderfully detailed page.
It makes me wonder if a formal naming process is also possible, given that origami is a largely linear, and well defined process.
You'd start with a prefix which denotes the shape of the paper (square or aspect ratio). Then lists the required transforms in order. There would be some ambiguity as order of certain transformations might not be critical. Perhaps you could develop standard notation there.
I see that there appears to be some academic software for modeling origami structures. So I assume this problem must have been addressed to some extent: