Let $p\in [1,\infty]$ and let $X$ be a separable Banach space. Then $X$ is said to be a $\mathscr{L}_p$-space if there exists an increasing union of finite dimensional Banach spaces $F_n\subset F_{n+1}\subset \ldots$ with union dense in $X$ such that for some $\lambda>0$ the Banach-Mazur distance satisfies $d(F_n, \ell_p^{\dim F_n}) \leqslant \lambda$ for all $n$.

Many authors use without explanation the following definition:

$X$ is a $\mathscr{L}_p$-space if there exists an increasing net $(F_\alpha)$ of finite-dimensional subspaces with $X=\bigcup_\alpha F_\alpha$ such that for some $\lambda>0$ the Banach-Mazur distance satisfies $d(F_\alpha, \ell_p^{\dim F_\alpha}) \leqslant \lambda$ for all $\alpha$.

Why do these definitions turn out to be equivalent?

Secondly, is there anything special about the choice of $\ell_p$ here? Let us generalise this in the following manner.

Let $E_1 \subset E_2 \ldots $ be an increasing sequence of isometric embedding of finite-dimensional Banach spaces. Let $E$ be the direct limit of this sequence. Let us sat that $X$ is an $\mathscr{L}_E$-space if there exists an increasing net $(F_\alpha)$ of finite-dimensional subspaces with $X=\bigcup_\alpha F_\alpha$ such that for some $\lambda>0$ the Banach-Mazur distance satisfies $d(F_\alpha, E_{\dim F_\alpha}) \leqslant \lambda$ for all $\alpha$.

Suppose that $X$ is a separable Banach space. Suppose moreover that there exists an increasing union of finite dimensional Banach spaces $F_n\subset F_{n+1}\subset \ldots$ with dense union such that for some $\lambda>0$ the Banach-Mazur distance satisfies $d(F_n, E_{\dim F_n}) \leqslant \lambda$ for all $n$.

Is $X$ a $\mathscr{L}_E$-space?

If so,

Are ultrapowers of $\mathscr{L}_E$-spaces $\mathscr{L}_E$ as well?