I tried to write the explanation of the comment about a dozen of times, but was never satisfied with the result. Finally, I decided to write a full note (I will try to put in on arXiv in a few minutes) describing the natural setting for such questions:

http://www.mimuw.edu.pl/~mrp/the_other_pullback_lemma.pdf

(the note still needs some improvements, but I am running out of time now...)

I found it easier to characterise your condition by "extremal epimorphisms" rather than "strong epimorphisms" (notice however, that in case of finite connected limits, these concepts coincide). Here is the formal statement:

Let us assume that finite connected limits exist. The following are equivalent:

• your condition along $e \colon X \rightarrow Y$ holds
• $e \colon X \rightarrow Y$ is an extremal morphism stable under pullbacks.

I tried to write the explanation of the comment about a dozen of times, but was never satisfied with the result. Finally, I decided to write a full note (I will try to put it on arXiv in a few minutes; here it is) describing the natural setting for such questions:

http://www.mimuw.edu.pl/~mrp/the_other_pullback_lemma.pdf

(the note still needs some improvements, but I am running out of time now...)

I found it easier to characterise your condition by "extremal epimorphisms" rather than "strong epimorphisms" (notice however, that in case of finite connected limits, these concepts coincide). Here is the formal statement:

Let us assume that finite connected limits exist. The following are equivalent:

• your condition along $e \colon X \rightarrow Y$ holds
• $e \colon X \rightarrow Y$ is an extremal morphism stable under pullbacks.