Okay, you know how film adaptions are always different from the book, because they are translations from text to visual media, and things are lost and added and changed? Well, it's an absolutely terrible metaphor, but it's the best way I can think off to describe how the codification of a spell works.
So will the codified version of Rite of Way be different from Mathilde's version? Hopefully not, but we won't know until we try it, and that could take several rounds of trial and error casting, which might be dangerous for bystanders, which links back to the original point of maybe we shouldn't have Eike standing nearby as we experiment with battle magic.