The condition that pairs of disjoint open sets have disjoint closures is actually equivalent to extremal disconnection, not basic disconnection. (Also note that “$^*$extremely disconnected” should be replaced with “extremally disconnected.”)
Necessity$\phantom{--}$ Suppose that $X$ is extremally disconnected. Let $U,V\subseteq X$ be disjoint open sets. For the sake of contradiction, assume that $x\in(\operatorname{cl} U)\cap(\operatorname{cl}V)$. Since $V^{\mathsf c}$ is closed and $U\subseteq V^{\mathsf c}$, it follows that $\operatorname{cl}U\subseteq V^{\mathsf c}$. Similarly, $\operatorname{cl} V\subseteq U^{\mathsf c}$. Therefore, $x$ is neither in $U$ nor in $V$. But since $x\in\operatorname{cl} U$, it must be an accumulation point of $U$. Given that $\operatorname{cl} V$ is an open neighborhood of $x$, there must exist some $y\in U\cap(\operatorname{cl} V)\subseteq U\cap U^{\mathsf c}=\varnothing$, a contradiction.
Sufficiency$\phantom{--}$ For the other direction, suppose that $X$ is not extremally disconnected. Then, there exists an open set $S\subseteq X$ such that $\operatorname{cl} S$ is not open. This means that $\operatorname{int}(\operatorname{cl}S)$ is a strict subset of $\operatorname{cl}S$, so that there exists some $x\in\operatorname{cl}S$ such that $x\notin \operatorname{int}(\operatorname{cl}S)$. But then $$x\in[\operatorname{int}(\operatorname{cl}S)]^{\mathsf c}=\operatorname{cl}[\operatorname{int}(S^{\mathsf c})],$$ so that $S$ and $\operatorname{int}(S^{\mathsf c})$ are disjoint open sets, yet their closures intersect.
