Bas
Interesting topic, for sure. I've thought about this very concept in my own teaching and I've come to the conclusion that there's so much variation in string grips to be standardized, at least in my teaching system, that the memorization capacity of the student would be taxed more than necessary. For me, naming the grips is be less efficient than simply calling off the strings to be plucked.
As an example, the basic strings for E9 grips using pedals A, B, and the F lever, as we know, are 10, 8, 6, 5 4, 3. So within that group, there are the closed voicing grips, which could be Grip 1 (345), Grip 2 (456), Grip 3 (568), and Grip 4 (6810).
Then there are the open voicing grips, which could be Grip 5 (358), and Grip 6 (4610)
Bringing string 2 and 7 into the mix, as when the E lever is being used, the grips then might become Grip 7 (125), Grip 8 (257), Grip 9 (457), Grip 10 (578), Grip 11 (7810).
We're at eleven grips now and that's just for major chords; minors, dominants, and the various extensions have yet to be addressed. That's a lot to be memorized for a student not yet familiar with all the possible combinations of strings/pedals/levers.
I usually just say "hit 3-5-8, with the E lower lever," or some such designation.
I also eschew

using "lever E," "lever F" et al., and prefer using "E lower lever," "E raise lever," etc., since it specifically tells the student what the lever does. Lack of standardization is the problem. For example, Buddy Emmons' nomenclature for levers is different than Jeff Newman's. Using "raise" and "lower" gives more specificity.
This is just my teaching system, of course. There are many ways to skin that cat, as we all are, or should be, aware.
In a handout sheet I give to my students... using only pedals ABC and levers DEF... there are 20 grips for a major chord, 17 for a minor chord, and 22 for a dominant chord.
My rig: Infinity and Telonics.
Son, we live in a world with walls, and those walls have to be guarded by men with steel guitars. Who's gonna do it? You? You, Lt. Weinberg?