The second statement follows from the first since divisorial points are dense in $X$, see §2.1. To prove the first statement, pick an SNC model $\mathcal{X}$ of $X$ such that $x=x_{E}$ is associated to an irreducible component $E_{\omega}$ of the special fiber. By [BFJ11, Proposition 5.2] there exists $u\in\mathcal{D}(X)$ determined on $\mathcal{X}$ such that $-1\leq u\leq 0$ and $\omega+dd^{c}u$ is determined by an ample class in $N^{1}(\mathcal{X}/S)$. Then $\Capa_{\omega}\{x\}\geq\MA(u)\{x\}=b_{E}((\omega+dd^{c}u)|_{E})^{n}>0$, see §2.7. ∎