Pitching change ?

[MTR]

Local rule?

Here's a thought. No one is really a pitcher is a particular inning until they have actually started a pitch per 6.2/Fastpitch "the pitch starts when one hand is taken off the ball after the hands have been placed together." (ASA 2006. Sorry did not have 2009 available).

Well, that isn't exactly factual, but it could be used as an indicator. The pitcher is whoever the coach says is the pitcher.

[HugoTafurst]

Seems the answer is in the definition of a substitute - in particular, an unannounced substitute:
The NFHS definition is as follows:

Should there be no announcement of substitutions,
a substitute has entered the game when the ball is live and:
a. a runner takes the place of a runner she has replaced.
b. a pitcher takes her place on the pitcher's plate.
c. a fielder reaches the position usually occupied by the fielder she has
replaced.
d. a batter takes her place in the batter's box.
e. and, in each of the above situations, when the ball is declared live by the
plate umpire.

"When the ball is live" is the key here.