I can think of only two actual coordinate systems for this. One would be to number each of the eight cubes, and designate each face according to which two cubes it connects The other is to assign a binary notatioin, with White's Home Face as 0000 and Black's as 1111.
I think you're forgetting the one I already described. You were able to tell from my description of it that it accurately represented the geometry of a tesseract. Additionally, I was able to describe it clearly without using any labeled diagrams. The only image it referenced was the 2D representation of a tesseract that looks like one cube inside of another with lines connecting their corners. It works like this:
Initially number the inner and outer cubes like a die with 1 opposite 6, 2 opposite 5, and 3 opposite 4, using the same numbers for the same sides of each.
Designate a face on the inner cube by preceding it with 0, which gives us 01, 02, 03, 04, 05, and 06.
Designate a face on the outer cube by preceding its number with the same number, which gives us 11, 22, 33, 44, 55, and 66.
Designate any face between the inner and outer cubes by the two numbers initially given to the inner and outer cube faces it lies between, writing the lower number first to avoid redundancy. This gives us 12, 13, 14, 15, 23, 24, 26, 35, 36, 45, 46, and 56.
I think you're forgetting the one I already described. You were able to tell from my description of it that it accurately represented the geometry of a tesseract. Additionally, I was able to describe it clearly without using any labeled diagrams. The only image it referenced was the 2D representation of a tesseract that looks like one cube inside of another with lines connecting their corners. It works like this: