We start by proving the first equality. Let $x\in\mathbb{T}^{\an}$, we know from Lemma 1.5.3 below that $x$ has a center on $\mathscr{X}$ if and only if $\zeta(\val(x))$ has a center on $\mathscr{X}$. Thus, it is enough to prove that for $n\in N_{\mathbb{R}}$ and $y=\zeta(n)$, $y$ has a center on $\mathscr{X}$ if and only if $n\in\lvert\Sigma_{1}\rvert$.
The elements $y\in\zeta(N_{\mathbb{R}})$ are precisely the valuations invariant under the torus action, hence if $y$ has a center on $\mathscr{X}$, it must be the closure of a torus orbit $Y\subset\mathscr{X}_{k}$. By [KKMSD73, Theorem 6], there exists a cone $\sigma\in\hat{\Sigma}$ such that the generic point of $Y$ is contained in the associated toric affine chart $\mathscr{X}_{\sigma}=\Spec R[\check{\sigma}\cap\hat{M}]$. In particular, for any monomial $z^{m}$ that is regular on $\mathscr{X}_{\sigma}$, we have $v_{y}(z^{m})\geq 0$. In other words, writing $y=\zeta(n)$, we have $\langle n,m\rangle\geq 0$ for all $m\in\check{\sigma}$, so that $n\in\sigma$. Since $v_{y}(t)=1$, $y\in\zeta(\lvert\Sigma_{1}\rvert)$.
By the same argument, if $n\in\lvert\Sigma_{1}\rvert$, there exists a cone $\sigma$ such that $n\in\sigma$, which means that $v_{\zeta(n)}$ has positive value on each monomial $m\in\check{\sigma}$, and thus has a center on $\mathscr{X}_{\sigma}$ and in particular on $\mathscr{X}$.

To prove the second equality, since $\rho_{\mathscr{X}}$ is the identity on $\zeta(\lvert\Sigma_{1}\rvert)=\Sk(\mathscr{X})$, we merely have to prove that $\rho_{\mathscr{X}}=\rho_{\mathscr{X}}\circ\val$. However this follows directly from the definition of $\rho_{\mathscr{X}}$, and the fact that $c_{{\mathscr{X}}}(x)\in\overline{c_{{\mathscr{X}}}(\zeta(\val(x)))}$ for $x\in\widehat{\mathscr{X}}_{\eta}$ by Lemma 1.5.3. Indeed, $\rho_{\mathscr{X}}(x)$ only depends on the values $v_{x}(z)$, where $z$ is a local equation for a component of $\mathscr{X}_{k}$ at $c_{\mathscr{X}}(x)$. Since $\mathscr{X}$ is a toric model, these local equations can be taken to be monomials, so that the result follows from the fact that $x$ and $\zeta(\val(x))$ take the same values on monomials. ∎