Let $R$ be Noetherian local ring of dimension $d$ with maximal ideal $m$, and $k=R/m$. There are (at least) two definitions of a dualizing (canonical) module over $R$.
In the first definition $D$ is dualizing if it is finitely generated maximal Cohen-Macaulay module of finite injective dimension and type one.
In the second definition $D$ is finitely generated module and for top local cohomology of $R$ there is an isomorphism $H^d_m(R) \cong \operatorname{Hom}(M, E(k))$, where $E(k)$ is the injective hull of $k$.
From the second definition it is easy to derive the local duality theorem, then from local duality we see that $\operatorname{Ext}^i(k, D)=\delta_{i,d} k$, which is the first definition.
Is there a direct (avoiding local duality) proof that the first definition implies the second?