The term 'extension' could generally be interpreted in mathematical sense1 as an enlargement of the domain of a pre-existing function.
In your case, assume Density-reachability could be described as a process $d\colon A \rightarrow B$ mapping inputs from domain $A$ to output in domain $B$. An enlargement of domain could be viewed as another map $e\colon A \rightarrow A'$ which maps where the old inputs in $A$ could be thought of as special cases in subset of $A'$ that is the image of map $e$.
An extension then could be described as any map $x\colon A' \rightarrow B$ which statisfy
$$x \circ e = d$$
There could be many such $x$, so picking extension is a choice. The above equation basically expresses that
If I pick $a \in A$ from the old domain, translate it to the new domain as $e(a)$, then use this new tool $x$ which could be thought of as an extension of old tool $d$ along the translation $e$, then the final result must agree with just using the old tool $d$ to begin with.
$$x \circ e(a) = d(a)\qquad; \forall a \in A$$
The term 'canonical' simply implies that the choice of extension is the obvious, natural one of the situation.
1 At least from the perspective of category theory.