It would seem that there are two needs immediately visible.

One need is for a notation to communicate a metamorphosis of the cube
from one cube-lover to another hexahedrophile. This notation should
capture the spirit of the thing being described; it should be rich and
chunky (like soup?).

The other need is for an archival notation, for reference and
cataloguing. It must be spare and canonical. There must be one and only
one way to describe a sequence of moves. (Implication: a way to get
reflections, inversions etc. out of the notation? how?) No nourishment
for the spirit there, but when you need to look something up...

Well, clearly they can't be the same notation. (Though even the archival
notation could invoke functions.) Or can they? Food for thought...

Anyway, plain ole BFUDLR can't do either job decently. Not rich or
expressive enough for the first need, too expressive for the second
(gee, I wonder how many ways there are to annotate PONS?)

Well, other people can muddle this out - I'm going home to get some
dinner.