When does player officially leave the game
 

Team has 9 players to start the game. Player gets hurt as BR when she is called out. She does not play on defense for 2 innings. When her next time at bat comes up, she bats. Since the team only had 9 players, she was never replaced by a substitute and the team played defense with 8 players.

Question: Did she officially leave the game when she got hurt and re-entered to bat, or was she still officially in the game even though she did not play defense? Remember, she never missed a time at bat.

Speaking ASA, I think you're looking for something that isn't there. She leaves the game when she ceases to play her position or bat in her spot.

The team played shorthanded for 2 innings. The player who leaves the game under the shorthanded rule may not reenter. ASA 4-1-D-2-f.

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.

NCAA it is a forfeit under rule 8.1.1.

For ASA and NFHS, Both rule sets defined a shorthanded situation as a team that cannot provide the required number of players in the batting order. There is no reference to fielding. Therefore, I would not invoke the shorthanded rule until such time as the player misses a turn at bat. (ASA - 4.1.d.2.a; NFHS - 4-3-1-g)

Also to be clear that the penalty for an illegal player uner ASA is a disqualification of the player under rule 4.6.E - Effect)

Was there a substitute available?

I disagree. 4-1-d-2-d refers to "if" the player is a runner or batter... leaving the possiblity of them NOT being a runner or batter (e,g, a fielder). Reference to the batting order is for count purposes (when do you go shorthanded) and for charging the out (... 2-e), which, BTW, also implies they may have been playing shorthanded for some time before the player is due to bat.

I got to learn this rule first hand at a Women's ASA FP National. Exact scenario you cited happened at my game. Player got overheated and left game, no subs available. About two innings later she came back and I allowed it. Found out after the game, from the UIC, that I shouldn't have allowed it (made no difference in the game as the shorthanded team was getting it handed to them anyway, and opposing team was happy she came back). Once they took field and played without her, she had left the game. Rules do not permit her to come back.

Personally, I don't understand why the player isn't allowed to come back. I'm sure the rule serves to protect some advantage but don't see what it is.

What about all the defense being required to be in fair ground, exc. F2?

What does "invoke the shorthanded rule " imply other than when a team is at bat?

Remember the alternative prior to the shorthanded rule. The idea of the rule was to avoid forfeits. The team is already being given a break by being allowed to continue play as opposed to packing their bags.

Since a team can start or continue play with less than the "required" number of players without qualifying a cause, I guess there could be an abuse of the rule.

Simple. Goes like this...

"Oh coach, I'm not batting so well today."

"No problem, Mary. Fake an injury and we'll go shorthanded. After your batting spot is skipped, we'll put you back on defense in this tight game."

"Thanks, coach! You're swell!"

Okay, cheesy, I know. But don't think for a moment that a coach wouldn't pull that if given the option.

But that would result in an out. The advantage to not playing defense is the question.
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Assume runners on 1st and 3rd, 1 out. B9 due up, tends to GIDP. B1 follows who has a 500 OBP and a 400 BA. Coaches play the odds. But, it would be simpler to just give B9 the "take" sign on every pitch...

How's this for a philosophy?

If she isn't physically capable to stand in one spot on defense, she probably shouldn't be batting and running the bases, either.

Or,

You can't have a DP without the FLEX.

"Oh, coach, it's so hot. And, if I have to field a ball, I might break a nail."

"Mary, we would forfeit if you don't play. I know you like to hit, so how about if we just say you are injured, and you can hit so we don't forfeit." Meanwhile, Coach is thinking "She isn't any help on defense, anyway; always thinking she might break a nail if she actually had to field the ball."