The game-theoretical terminology is quite confusing. At the original Shapley paper, the symmetry axiom means that the Shapley value is invariant with respect to any players' permutation, i.e. $$ \Phi _{\pi i}( \pi \cdot v)=\Phi _i(v) $$ But at the many modern texts, including textbooks, the symmetry means different axiom, something like "mirroring"
If $v (S\cup \{i\})=v (S\cup \{j\})$ for any coalition $S: i,j\notin S$, then $\Phi _i(v)=\Phi _j(v)$
and the original symmetry axiom is now called anonymity.
Which of them terminologies is clearer in today's game theory? How to avoid confusion? I take care of this article on Wikipedia.