Proof.
We first show that $(1) \Rightarrow (2)$. Assume that $\overline{W}$ is accessibly strongly saturated: that is, it is generated (as a strongly saturated collection of morphisms) by a small subset $W \subseteq \overline{W}$. As in the proof of Theorem 9.6.9.8, we can arrange that $W$ is closed under the formation of relative codiagonals. It follows that an object of $\operatorname{\mathcal{C}}$ is $W$-local if and only if it is weakly $W$-local (Proposition 9.6.2.14).
Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. It follows from Theorem 9.6.9.8 that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. By construction, every element of $W$ is a $\operatorname{\mathcal{C}}_0$-local equivalence. Since the collection of $\operatorname{\mathcal{C}}_0$-local equivalences is strongly saturated (Example 9.6.9.21), it follows that every element of $\overline{W}$ is a $\operatorname{\mathcal{C}}_0$-local equivalence. For every object $X \in \operatorname{\mathcal{C}}$, Theorem 9.6.3.3 guarantees that there is a morphism $f: X \rightarrow Y$, where $Y \in \operatorname{\mathcal{C}}_0$ and $f$ is a (small) transfinite pushout of morphisms of $W$, and therefore belongs to $\overline{W}$. Since $\overline{W}$ has the two-out-of-three property, Lemma 6.2.4.1 guarantees that $\overline{W}$ is the collection of all $\operatorname{\mathcal{C}}_0$-local equivalences in $\operatorname{\mathcal{C}}$.
We now complete the proof by showing that $(2) \Rightarrow (3)$ (the implication $(3) \Rightarrow (1)$ is trivial). Suppose that $\overline{W}$ is the collection of $\operatorname{\mathcal{C}}_0$-local equivalences, where $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. Invoking Example 9.6.9.21, we see that $\overline{W}$ is strongly saturated. Let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a left adjoint to the inclusion functor and let $\mathcal{W}$ be the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the morphisms of $\operatorname{\mathcal{C}}$ which belong $\overline{W}$. Then we have a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{W}}\ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{L \circ } \\ \operatorname{Isom}( \operatorname{\mathcal{C}}_0 ) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_0 ). } \]
Applying Corollary 9.5.4.6, we conclude that the $\infty $-category $\operatorname{\mathcal{W}}$ is presentable (and closed under colimits in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}$), so that $\overline{W}$ is generated by a small subset under (small) colimits.
$\square$