13-move miniature checkmate with the Advance French

Up to 5...Qb6 is mainline theory afaik, but White deviates with 6. Ne5?!. After 6...Nxe5 7. dxe5 all White has accomplished is to ruin his own pawn structure. The E5 pawn does prevent ...Nf6, but my knight would be going to E7 in this structure anyway.

9. Qe2 aims to exchange queens, eg. 10. Qb5+ Bd7 11. Qxb6 axb6, but 9...c4 stoppers up the battery -- and, importantly, exposes the E3 pawn. And the king is just behind the pawn!

10. Qd1?? (or just Qd1?, depending on your opinion) falls straight into my attack, which begins with 10...Qxe3+. White not only lost the pawn with check, but also can no longer castle. 11. Be2?! should be Qe2, after which if I want to avoid a queen exchange, I have to retreat.

With 11...Bc5, checkmate in one is threatened with ...Qf2#. The engine suggests 12. Nf3 (...Qf2+ 13. Kd2), with the king running to the queenside. With 12. Rf1?! however, White's kingside resources are running thin.

This isn't a perfect miniature since after 13. Rxf4 Qxf4 14. Nf4! White is going to survive a while further into the game. Happily (for me), 13. Nf3??? allows a rather pretty mate.

French with 5 e3?

Yeah, in the French Defence Advance Variation, the white e-pawn should be on e5.
And that's also why Qb6 is less strong in this position than in the French Defence: you are pressuring the d4 pawn that is well defended by pawns on e3 an c3.
But nice game from your side.

And c4 gives away the d4 square. If I was white, I would have planted my knight there, Nd2-f3-d4.

Yeah, it's not a perfect miniature.

Can someone define a miniature, to me that game just looked like a blunder fest for white

A miniature is a game shorter than 25 moves, iirc.