The equivalence of (1) and (2) is the definition of (1). The equivalence of (2), (3), and (4) is Lemma \ref{lemma-unramified-in-L}. It is straightforward to prove that (4) $\Rightarrow$ (5), (5) $\Rightarrow$ (6), (6) $\Rightarrow$ (7). \medskip\noindent Assume (7). Observe that $\mathcal{O}_{X, x}^{sh}$ is a normal local domain (More on Algebra, Lemma \ref{more-algebra-lemma-henselization-normal}). Let $L^{sh} = L \otimes_K K_x^{sh}$ where $K_x^{sh}$ is the fraction field of $\mathcal{O}_{X, x}^{sh}$. Then $L^{sh} = \prod_{i = 1, \ldots, n} L_i$ with $L_i/K_x^{sh}$ finite separable. By Algebra, Lemma \ref{algebra-lemma-integral-closure-commutes-smooth} (and a limit argument we omit) we see that $Y \times_X \Spec(\mathcal{O}_{X, x}^{sh})$ is the integral closure of $\Spec(\mathcal{O}_{X, x}^{sh})$ in $L^{sh}$. Hence by Lemma \ref{lemma-unramified-in-L} (applied to the factors $L_i$ of $L^{sh}$) we see that $Y \times_X \Spec(\mathcal{O}_{X, x}^{sh}) \to \Spec(\mathcal{O}_{X, x}^{sh})$ is finite \'etale. Looking at the generic point we see that the degree is equal to $[L : K]$ and hence we see that (8) is true. \medskip\noindent Assume (8). Assume that $x \in X$ and that the scheme theoretic fibre $Y_x$ is \'etale over $x$ of degree $\geq [L : K]$. Observe that this means that $Y$ has $\geq [L : K]$ geometric points lying over $x$. We will show that $Y \to X$ is finite \'etale over a neighbourhood of $x$. This will prove (1) holds. To prove this we may assume $X = \Spec(R)$, the point $x$ corresponds to the prime $\mathfrak p \subset R$, and $Y = \Spec(S)$. We apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-makes-integral-split} and we find an \'etale neighbourhood $(U, u) \to (X, x)$ such that $Y \times_X U = V_1 \amalg \ldots \amalg V_m$ such that $V_i$ has a unique point $v_i$ lying over $u$ with $\kappa(v_i)/\kappa(u)$ purely inseparable. Shrinking $U$ if necessary we may assume $U$ is a normal integral scheme with generic point $\xi$ (use Descent, Lemmas \ref{descent-lemma-locally-finite-nr-irred-local-fppf} and \ref{descent-lemma-normal-local-smooth} and Properties, Lemma \ref{properties-lemma-normal-locally-finite-nr-irreducibles}). By our remark on geometric points we see that $m \geq [L : K]$. On the other hand, by More on Morphisms, Lemma \ref{more-morphisms-lemma-normalization-smooth-localization} we see that $\coprod V_i \to U$ is the normalization of $U$ in $\Spec(L) \times_X U$. As $K \subset \kappa(\xi)$ is finite separable, we can write $\Spec(L) \times_X U = \Spec(\prod_{i = 1, \ldots, n} L_i)$ with $L_i/\kappa(\xi)$ finite and $[L : K] = \sum [L_i : \kappa(\xi)]$. Since $V_j$ is nonempty for each $j$ and $m \geq [L : K]$ we conclude that $m = n$ and $[L_i : \kappa(\xi)] = 1$ for all $i$. Then $V_j \to U$ is an isomorphism in particular \'etale, hence $Y \times_X U \to U$ is \'etale. By Descent, Lemma \ref{descent-lemma-descending-property-etale} we conclude that $Y \to X$ is \'etale over the image of $U \to X$ (an open neighbourhood of $x$). \medskip\noindent Assume $L/K$ is Galois and (9) holds. Then $Y \to X$ is \'etale by Lemma \ref{lemma-inertial-invariants-unramified}. We omit the proof that (1) implies (9).