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Field of definition of local fields of positive characteristic

Let $K$ be a number field and let $L/K$ be a finite Galois extension. I’d like to know, when is it true that $K^{\mathrm{sep}}$ is the field of definition of $L/K$?
This is clearly true when $L/K$ is a Galois extension of prime-to-$p$ degree. (this is just Galois theory: if $A,B$ are normal, local, DVR with uniformizing parameter $u$ and if $\sigma\in \mathrm{Gal}(L/K)$, then $A^{\sigma}=A[\frac{u}{1+\sigma(u)}]$ and $B^{\sigma}=B[\frac{u}{1+\sigma(u)}]$).
It is also true for $p$-adic Lie extensions (applying that $p$ is odd and $L$ is local).
Edit: I learned from a nice comment by Roquette that in general a finite Galois extension is separable (hence, assuming the separability hypothesis, this is true). I was actually confused by a remark about $\mathbb{Q}_{p}$ vs $\mathbb{Q}_{p}(\mu_{p})$, saying it was always defined over $\mathbb{Q}_{p}$, when indeed, it’s separable.
Let $L/K$ be a finite separable extension of algebraically closed fields with a separable closure $K^{\mathrm{sep}}/K$. Is it true that $K^{\mathrm{sep}}$ is the field of definition of $L/K$? I guess that this is true if $L/K$ is the splitting field of a polynomial with coefficients in $K$ (since the rational functions are algebraically closed in this case), but it is clearly false if $L/K$ is a finite extension of $\mathbb{Q}_{p}$ (think of the extension of

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