When does a Player assume pitching duties

[Rattlehead]

Defensive team is entering the field, Third Baseman picks up game ball and starts goofing off with the ball in the circle. She even acts like she is going to pitch the ball but does not ever throw the ball to the catcher. She then hands the ball the pitcher and then starts to third.

PU then says that she must deliver one pitch because the player is now the pitcher.

I looked (Skimmed)for it but I couldn't find the rule.

Doesn't the player have to throw a warmup pitch to be considered a pitcher sub?