Let $X$ be a locally noetherian adic space (for example, $\operatorname{Spa(K,\mathcal{O}_K)}$). In Scholze's paper "$p$-adic hodge theory for rigid-analytic varieties ", he defined pro-étale map in the following way: A morphism $U\rightarrow V$ of objects of pro$-X_{ét}$ is called étale if $U=U_0\times_{V_0}V$ via some étale morphism $U_0\rightarrow V_0$ of objects in $X_{ét}$.
A morphism $U\rightarrow V$ of objects of pro$-X_{ét}$ is called $\textbf{pro-étale}$ if it can be written as a cofiltered limit $U=\varprojlim U_i$ of objects $U_i\rightarrow V$ étale over $V$, such that $U_i\rightarrow U_j$ is finite étale and surjective for large $i>j$.
Use such definition, he verified that any pro-étale map $U\rightarrow V$ in pro$-X_{ét}$ is open.
In his Berkeley notes, he gave another definition for pro-étale maps of perfectoid spaces. A morphism $f: \operatorname{Spa}\left(B, B^{+}\right) \rightarrow \operatorname{Spa}\left(A, A^{+}\right)$ of affinoid perfectoid spaces is callled (affinoid)$\textbf{pro-étale}$ if
$\left(B, B^{+}\right)=\varinjlim \widehat{\left(A_{i}, A_{i}^{+}\right)}$
such that $\operatorname{Spa}\left(A_{i}, A_{i}^{+}\right) \rightarrow \operatorname{Spa}\left(A, A^{+}\right)$ is étale. A morphism of perfectoid spaces is $\textbf{pro-étale}$ if it is locally affinoid pro-étale.
Under this second definition, pro-étale morphisms are not necessarily open and he gave a counterexample:
$\operatorname{Spa}\left(K, \mathcal{O}_{K}\right) \times\{x\}\rightarrow \operatorname{Spa}\left(K, \mathcal{O}_{K}\right) \times \underline{S}$
for a profinite set $S$ by considering $x$ as a intersection of open and closed subsets of $S$.
I guess that that there must be some difference between such two definitions when restricted to perfectoid spaces over $\operatorname{Spa(K,\mathcal{O}_K)}$. It seems the first definition is stronger since we require surjectivity of transition maps. Moreover, I think the map in this counterexample is not considrered as pro-étale using the first definition. A hint is otherwise we would get a contradiction (Using first definition any pro-étale map $U\rightarrow V$ in pro$-X_{ét}$ is open). Is my sense correct?