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For a finite group $G$ with a finite symmetric generating set $S$, its diameter is the maximum word metric distance between two elements, equivalently the diameter of its Cayley graph. PutThenTranslation invariance reduces the maximum pairwise distance to the maximum distance from the identity, which proves the equality.

If $1\in S$, one may replace $B_S(n)$ by $S^n$. Otherwise exact powers may never equal $G$: for $G=C_2=\{1,a\}$ and $S=\{a\}$, even powers are $\{1\}$ and odd powers are $\{a\}$, while the actual Cayley graph has diameter one. This is why the ball convention matters. The trivial diameter bound for a finite group gives ${\mathrm{diam}}_S(G)\le |G|-1$.