of a given category$\mathcal{D}$ is meant to be full, if it includes “some objects but all the morphisms between these objects”.

This means at least that $\iota$ is a fully faithful functor. In fact, that is the most one may demand while respecting the principle of equivalence of category theory and hence constitutes an invariant definition of full subcategory (Def. below).

If one accepts the notion of subcategory without any qualification (as discussed there), then:

A subcategory$S$ of a category $C$ is a full subcategory if for any $x$ and $y$ in $S$, every morphism $f : x \to y$ in $C$ is also in $S$ (that is, the inclusion functor$S \hookrightarrow C$ is full).

This inclusion functor is often called a full embedding or a full inclusion.

Notice that to specify a full subcategory $S$ of $C$, it is enough to say which objects belong to $S$. Then $S$ must consist of all morphisms whose source and target belong to $S$ (and no others). One speaks of the full subcategory on a given set of objects.