"Permutation of a set" vs "permutation of a Rubik's cube": are these uses of "permutation" equivalent? So my book defines a permutation as follows : "By a permutation of  a  set A  we  mean  a bijective  function  from A to A,  that  is,  a  one-to-one correspondence between A and itself."
This is perfectly clear to me.
However, when talking about objects like the Rubik's cube. People refer to the state of the Rubik's cube as a "permutation of the cube".
Now my question is: 

Are these definitions distinct? i.e., is "permutation" a homonym with two meanings, or are these definitions somehow equivalent?

 A: The answer ultimately depends on what you mean by "the state of the Rubik's cube."  I'll consider two cases, the first initially being more complicated, but for which the answer is easier, and the second being simpler, but with a more complicated answer.
Case 1: Take a solved rubik's cube (or similar, this will work for any size cube) and label each square with a distinct label and an orientation.  For instance, on a 3x3x3 cube, you can label each square on a given side with the numbers 1-9 and underline each number.  Finally, decide on an orientation for the colors of the solved cube (e.g. choose an orientation for the front, top, right corner).  In this case, we take the numbers and their orientations, as well as the orientation of our chosen corner, into account to define each state.
Case 2: Don't label the rubik's cube in any way.
Now apply a sequence of moves to the cube, writing down each as you go (e.g. I've seen people write "BBLRUDDDU..." for a 3x3x3, not sure what the notation is for larger cubes).  Each rotation itself is a permutation which we allow to act on the cube to yield a state of the cube, so the sequence of rotations is also a permutation.  In this way, we can define the permutations acting on the cube as any of the permutations you can get by a sequence of rotations.
As it was defined in Case 1, the resulting state of the cube can actually be identified by the product of rotations that reached that state from the "solved" state.  In this case, we can say that the given state of the cube represents the permutation we applied to the cube.  This allows us, for instance, to define a given state as a permutation you can then apply to any other state of the cube.
As it was defined in Case 2, the resulting state of the cube can come from many different permutations.  In fact, there are some permutations that would bring us back to a solved cube but would change an unsolved cube, so there's really no hope in identifying "permutations of a cube" with the underlying permutation applied to the cube.

That is, if you define "state," and therefore "permutation of the cube" using the definition given in case 1, then the permutations acting on the cube are in a natural correspondence with the states of the cube, so we can identify them.  
On the other hand, if you define "state" as in case 2, then the "permutations of the cube" are just the results of a given set of permutations acting on the cube. In general, however, there is no nice way to try to choose a permutation that the given state represents, so we have to conclude that defining "permutation of the cube" as the given state is a little abuse of notation.

A final note specific to the unlabeled 3x3x3 cube: it was pointed out to me that the orientation of the center cubes on each side can differ between two solved cubes (which is the reason I had to rewrite my original answer).  However, such a permutation, applied to an unlabeled cube, can never result in a different unlabeled state, so even though we still cannot identify the unlabeled state of the cube with the permutation itself, we can reasonably identify the state of the cube with an equivalence class of permutations.  It still is a little bit of an abuse of notation to identify the state as a permutation, but this shows that it's at least a harmless abuse of notation.
