Let $E/F$ be an algebraic field extension and $\bar F$ be an algebraic closure of $F$.
Define $[E:F]_{\text{sep}}$ as the cardinality of $$\{\sigma\in \operatorname{Mono}(E,\bar F): \sigma \text{ fixes } F\}$$ (it can be proven that this is well-defined)
With this terminology, here are two definitions for separable extension:
Definition 1
If every minimal polynomial $m$ of $\alpha\in E$ is separable, then $E/F$ is called separable.
Definition 2
If $[E:F]=[E:F]_{\text{sep}}$, then $E/F$ is called separable.
If $E/F$ is finite, then these two definitions coincide. However, I'm not really sure why Def. 1 is stronger than Def. 2 for infinite cases. Why Def. 1 is stronger? How do I prove that? And what is the standard one?