Assume (1). Then $Y$ is a scheme as a finite morphism is representable (by schemes), see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-local}. Hence (2) follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-surjective-finite-morphism-ample}. \medskip\noindent Assume (2). Let $P$ be the following property on coherent $\mathcal{O}_X$-modules $\mathcal{F}$: there exists an $n_0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. We will prove that $P$ holds for any coherent $\mathcal{O}_X$-module $\mathcal{F}$, which implies $\mathcal{L}$ is ample by Lemma \ref{lemma-vanshing-gives-ample}. We are going to apply Lemma \ref{lemma-property-higher-rank-cohomological}. Thus we have to verify (1), (2) and (3) of that lemma for $P$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since $H^p(X, -)$ is an additive functor. \medskip\noindent To see (3) let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. Let $i' : Z' \to Y$ and $f' : Z' \to Z$ be as in Lemma \ref{lemma-finite-morphism-Noetherian} and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma \ref{lemma-property-higher-rank-cohomological} which will finish the proof. Property (3)(a) we have seen in Lemma \ref{lemma-finite-morphism-Noetherian}. To see (3)(b) let $\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$. Denote $\mathcal{I}' \subset \mathcal{O}_{Z'}$ the quasi-coherent ideal $(f')^{-1}\mathcal{I} \mathcal{O}_{Z'}$, i.e., the image of $(f')^*\mathcal{I} \to \mathcal{O}_{Z'}$. By Lemma \ref{lemma-affine-morphism-projection-ideal} we have $f_*\mathcal{I}' = \mathcal{I} \mathcal{G}$. We claim the common value $\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'$ satisfies the condition expressed in (3)(b). First, it is clear that the support of $\mathcal{G}/\mathcal{G}'$ is contained in the support of $\mathcal{O}_Z/\mathcal{I}$ which is a proper subspace of $|Z|$ as $\mathcal{I}$ is a nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$. Recall that $f'_*$, $i_*$, and $i'_*$ transform coherent modules into coherent modules, see Lemmas \ref{lemma-finite-pushforward-coherent} and \ref{lemma-i-star-equivalence}. As $Y$ is a scheme and $\mathcal{L}$ is ample we see from Lemma \ref{lemma-vanshing-gives-ample} that there exists an $n_0$ such that $$ H^p(Y, i'_*\mathcal{I}' \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0 $$ for $n \geq n_0$ and $p > 0$. Now we get \begin{align*} H^p(X, i_*\mathcal{G}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) & = H^p(Z, \mathcal{G'} \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n}) \\ & = H^p(Z, f'_*\mathcal{I}' \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z, f'_*(\mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (f')^*i^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z, f'_*(\mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Z', \mathcal{I}' \otimes_{\mathcal{O}_{Z'}} (i')^*f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Y, i'_*\mathcal{I}' \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*} Here we have used the projection formula and the Leray spectral sequence (see Cohomology on Sites, Sections \ref{sites-cohomology-section-projection-formula} and \ref{sites-cohomology-section-leray}) and Lemma \ref{lemma-finite-higher-direct-image-zero}. This verifies property (3)(b) of Lemma \ref{lemma-property-higher-rank-cohomological} as desired.