A crease pattern [almost] defines an origami design so this is the closest to a formal description you can get. It's actually good enough for quite an advanced mathematical apparatus to have been developed which allows you to prove lots of things about origami in a strict, mathematical way. However, the crease pattern is still a 2D drawing, so while much simpler to describe than the final 3D shape, it's also too complex to be neatly described in words. You can, of course, spell out all the folds, their angles and lengths, but usually it will be too long to be called a name. For simplified cases with additional bounds, it is, however possible - in the post I list one such approach (by Goran Konjevod) which works for a certain type of tessellations.
A crease pattern can clearly be serialized, and the order of folding listed. What's wrong with a textual description that might take a couple of pages to print out? It seems to me that there's no real compression challenge here—the full description of any folding pattern can be expressed concisely.
Origami models aren't always solely described by their crease pattern. Even the traditional crane has a step where it gets inflated in a way that creases alone wouldn't capture.
Not saying it's intractable, just more nuanced than it first appears.
