Corroborating the answer already given with more precise terminology: Let $G$ be the free group on the letters $R$, $U$, $F$, $B$, $R$, $D$ (so, just words consisting of these letters and their formal inverses, with concatenation + reduction being the group operation), and let $S$ be the set of all valid cube states. $G$ acts on $S$ in the natural way, producing a Cayley graph (adjacent nodes being those achievable from one another by a single move). Then God's number is just the maximum path-distance away from the solved state, on this graph.
The question posed in the OP is: What if we consider God's number as being relative to a general state, rather than just the solved state? This is the same notion, since any one node on the Cayley graph can be carried to another one arbitrarily specified, via a graph automorphism (namely, by an appropriate permutation, i.e. element of $G$).
Altogether it's like the (lack of) difference between asking for the furthest distance from $(1,0)$ on the unit circle, versus the furthest distance between two arbitrary points on the unit circle; there is a family of symmetries to allow us to fix one of our parameters.