I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.

Let us fix a non-archimedean complete valuation field $k$ (the valuation is not necessarily non-trivial) The first definition is as in TEMKIN: An analytic space over $k$ consists of a locally Hausdorff topological space $X$, a net $\tau$ on $X$(regarded as a category), and a functor from $\tau$ to the category of $k$-affinoid spaces sending morphisms to affinoid domain embeddings, such that the underlying topological spaces of $U\subset X$ and $\tau(U)$ are naturally identified.

The second definition I found is in Berkovich's original book: A $k$-analytic space is a locally ringed space $X$ with an equivalence class of collections of pairs $(U_i,\phi_i)$, where $U_i\subset X$ is open, $\phi_i$ is an open immersion of $U_i$ into a $k$-affinoid space subject to the usual compatibility conditions as in the definition of a mainfold.

I'm not pretty sure what is the connection between these two definitions, I guess that the first definition axiomatizes analytic domains in the second definition, but I haven't been able to make this precise.