For example, Martin talked about the corner orbit, the edge orbit, and the
face center orbit of the 3x3x3....
David Singmaster, on the other hand, has always talked about the twelve
orbits of the constructable group of the 3x3x3, where orbits are defined
in terms of twists, flips, and parity....
When a group G has a representation as permutations of a set X, the
orbits are the equivalence classes of X induced by x~y if a (x)g=y for
some g in G. But these orbits will be different depending on the
representation, and in particular depending on X.
If we represent the Rubik group as the usual permutations on cubies
and facies, the orbits are corners, edges, etc. as Martin uses. I
agree this is the usual kind of orbit to talk about.
If we represent the Rubik group as permutations on itself (I think
it's called the right regular representation) you get one orbit. This
is always true of the right regular representation, since for any f, g
in G, let h=f'g, and we have (f)h = g, so f~g.
But consider the constructible group C, the set of positions you can
get by taking the cube apart and putting it back together. We can
extend the right regular representation to a representation on C. In
this case, there are twelve orbits of mutually accessible positions.
This is Singmaster's usage. They are indeed the cosets of C/G, as
with any subgroup of a larger group.
But the fact that we usually do not consider the group structure of C
(as in taking products of reassemblies) militates against calling them
cosets, so I can understand why Singmaster might prefer orbits. But
we have to remember to disambiguate which kind of orbit we are talking