The definition of Hausdorff measure $\mathcal{H}^s$ (for $s$ a non-negative real number) of a (Borel) measurable set $E \subset \mathbb{R}^d$ is well-known. It reads as $\mathcal{H} ^ s(E)=\lim_{\varepsilon \to 0^+} \mathcal{H} ^ s _\varepsilon (E)$, where $$ \mathcal{H} ^ s _\varepsilon (E) = \inf ~ \Big \{ 2^{-s} \sum_j (\mathrm{diam} F_j)^s : E \subset \bigcup _{j=1} ^\infty F_j , ~ \mathrm{diam} F_k < \varepsilon \Big \}.$$ My question is that, whether the definition of Hausdorff measure remains true, if one just consider covering balls $B=B(x,r) \subset \mathbb{R}^d$? This is similar to the definition of standard Lebesgue measure in $\mathbb{R}^d$.
In this case, one can propose an alternative definition of $ \mathcal{H} ^ s _\varepsilon (E)$ (if not wrong, and after some normalizations) as below: $$\inf ~ \Big \{ 2^{-s} \sum_j (\mathrm{diam} B_j)^s : E \subset \bigcup _{j=1} ^\infty B_j ,~~ B=B(x,r_j), x \in \mathbb{R}^d, r_j < \tfrac{1}{2}\varepsilon \Big \} \\ \approx \inf ~ \Big \{ \sum_j (r_j)^s : E \subset \bigcup _{j=1} ^\infty B_j ,~~ B=B(x,r_j), x \in \mathbb{R}^d, r_j < \tfrac{1}{2}\varepsilon \Big \}.$$ Are there examples, where the above definition of Hausdorff measure is not the same as the original definition?