When does player officially leave the game

[AtlUmpSteve]

Rules set is important to this response; in every case, the result is a forfeit, just different when (and if you had the opportunity be proactive and save the forfeit).

NCAA; when they didn't field 9 defensive players, game ended in a forfeit.

ASA; when they didn't field 9 defensive players, they were playing shorthanded. Unless you (umpire) invoked the blood rule, that player is no longer eligible to play (4.1-D(2)f), and her batting later was an illegal player. The penalty for her illegal re-entry is forfeit (4.8-A Effect).

NFHS; when they didn't field 9 defensive players, they were playing shorthanded (4-3-1g and 3-3-8d). Unless you (umpire) invoked the blood rule, that player is no longer eligible to play, and her batting later was an illegal substitute. The penalty for an illegal sub is disqualification, etc. (depending on when discovered; read rule 3-4 in its entirety). Most states would likely grant an administrative protest for using an ineligible player and grant a forfeit to the offended team.

In NCAA, the game was over immediately. In ASA and NFHS, you should have stopped the batter from illegally participating.