The two definitions are equivalent whenever quotients are stable under pullback and subobjects are stable under pushout, such as in a topos.

Just as with subobjects and quotient objects, we have that $X$ is a subquotient of itself, and subquotients of subquotients of $X$ are themselves subquotients of $X$ in a natural way.

Just as subobjects of a set $X$ are in correspondence with predicates on $X$ and quotients of $X$ are in correspondence with equivalence relations on $X$, subquotients of $X$ are in correspondence with partial equivalence relations on $X$.