Well, standard notation includes ways to annotate middle slices and wide turns. And standard notation doesn't care about colors. Not sure if that helps? https://jperm.net/3x3/moves
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I'm still learning cubing, but with my modded cube with dice dots. I'm no expert cuber (yet), but I'm looking to learn the cube plus document neat dice dot patterns on top of that.
My modded cube is backwards compatible, but the dice dots make it extra interesting within the subset of the combined puzzles.
Oh, one other detail about my modded cube..
Which side which color is on absolutely matters! Note that the only two edge pieces with dots/holes are on the yellow side (6). So the nifty dice dot patterns don't work right if you don't have the cube oriented right.
Not sure if I completely understand your question, and how this would affect dice patterns, but I think you're looking for the advanced notation here.
For rotations, just use x, y, & z moves from a fixed orientation (most people start with White on Top & Green in Front) or specify the center colors directly. Just mentioning the color on top isn't enough info btw.
Actually, for the valid dice dot patterns, yes indeed it's sufficient to just state the top color. The sequences of moves work in basically any orientation, as long as the top and bottom colors are known, the side colors can be pretty much anywhere.
The reason this basically works is that opposing sides of a cube are similar colors, white/yellow, blue/green, and red/orange. And the opposing sides of a die always add up to 7 - 1/6, 2/5, 3/4. And basically all the legit dice series pretty much exclusively uses x2 turns.
Yellow just happens to be the odd one out, as it's the only side that has 2 edge dots, all the rest of the dots are either in the centers or corners.
So basically, the most proper dice series pretty much always has either solids, crosses, or checkerboard sides, the only real catch is making sure not to end up with a 0/7 because of that goofy yellow side.
But even with that said, there's still way more convoluted permutations to the proper dice series and other nifty dot patterns to be found.