injective hull as the "topological dual to the completion"

Question

Dmitry Kaledin's notes (Russian) on cohomology of sheaves on algebraic varieties define (lecture 14) the injective hull of the residue field $k$ of a local ring $R$ as an injective module $I$ such that

1. its support is the maximal ideal $\mathbb{m}$
2. $\mathrm{Hom}(k, I) = k$

Question: is this definition equivalent to the standard one with essential injection (specialized to residue fields, of course)?

Further, the injective hull in this sense is explicitly described, if $R$ contains $k$, as the "topological dual of the completion" of $R$, $\varinjlim \mathrm{Hom}_k(R/\mathbb{m}^n, k)$ (note that the homomorphisms are taken over $k$!). While one can check that this is indeed an injective module, I wonder if it is an injective hull of $k=R/\mathbb{m}$ in the sense of the "essential injection definition".