Let $\mathcal{T}$ be a triangulated category. If $\mathcal{T}'$ is a triangulated subcategory, modern literature calls it a localising subcategory if $\mathcal{T}'$ is closed under taking $\mathcal{T}$-coproducts of its objects (assuming of course that $\mathcal{T}$-coproducts exist).
Let $\mathcal{A}$ is an abelian category and $K(\mathcal{A})$ is its homotopy category of complexes. Let $\mathcal{S}$ be the class of quasi-isomorphisms. Let $\mathcal{L}$ be a triangulated subcategory of $K(\mathcal{A})$ and let $\mathcal{S}'$ be the class of quasi-isomorphisms in $\mathcal{L}$. Then if $D(\mathcal{L})$ is the localisation of $\mathcal{L}$ at $\mathcal{S}'$ and $D(\mathcal{A})$ is the localisation of $K(\mathcal{A})$ at $\mathcal{S}$, then there is a canonical functor $F: D(\mathcal{L}) \rightarrow D(\mathcal{A})$.
A definition I have seen in older literature is that $\mathcal{L}$ is called a localising subcategory if the functor $F$ is fully faithful. The motivation for the name seemingly being that the functor "localises" to the subcategory.
Is there any connection between these two definitions? Does one imply the other, or is one equivalent to the other in the right situation?