You chose 9 0-0.
An excellent choice and the move Steinitz played in the game. Your king is safe and you are now threatening to win a piece on d5, so Black is forced to do something about it.
This game shows a theme which turns up quite often: one side leaves his king in the centre just too long, while the other side crashes through with the pieces and forces a checkmate.
Moral: do not swap the e-pawns before your king is safe.
9 ...              Be6
Choose White's next move.
I would play 10 Bg5
I would play 10 Nxd5
I would play 10 Qb3