Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

This map is usually called the Torelli map, not the Abel-Jacobi map. In any case, Mumford observed that a certain toroidal compactification of $\mathscr{A}_g$ admits an extension of the Torelli map; the original reference is this paper of Namikawa, I think. That paper doesn't give a very good moduli description of the map; luckily Alexeev does in this paper. I imagine everything can be made extremely concrete in genus 2, but I don't know a good reference for this.

In genus two the situation is very simple. All toroidal compactifications of $A_2$ are isomorphic and the DM compactification $\overline M_2$ is a toroidal compactification. I don't know a good reference unfortunately.