At what point does a pitcher become the pitcher? Warm-ups, first batter, when? Can't find it in any rule book.

Once he assumes the legal position. Pretty much like an unreported, legal sub.

OBR - when he toes the rubber

LL - when he toes the rubber AND throws one warm-up.

FED - ?????

(For all these, I assume you're asking about unannounced subs)

FED -- when the ball becomes live.

If it's the pitcher starting the game, he's the official pitcher when the batting orders are given to the umpire in chief.

3.05
a) The pitcher named in the batting order handed the umpire in chief, as provided in Rules 4.01 (a) and 4.01 (b), shall pitch to the first batter or any substitute batter until such batter is put out or reaches first base, unless the pitcher sustains injury or illness which, in the judgment of the umpire in chief, incapacitates him from pitching. (b) If the pitcher is replaced, the substitute pitcher shall pitch to the batter then at bat, or any substitute batter, until such batter is put out or reaches first base, or until the offensive team is put out, unless the substitute pitcher sustains injury or illness which, in the umpire in chief's judgment, incapacitates him for further play as a pitcher. (c) If an improper substitution is made for the pitcher, the umpire shall direct the proper pitcher to return to the game until the provisions of this rule are fulfilled. If the improper pitcher is permitted to pitch, any play that results is legal. The improper pitcher becomes the proper pitcher as soon as he makes his first pitch to the batter, or as soon as any runner is put out. If a manager attempts to remove a pitcher in violation of Rule 3.05 (c) the umpire shall notify the manager of the offending club that it cannot be done. If, by chance, the umpire in chief has, through oversight, announced the incoming improper pitcher, he should still correct the situation before the improper pitcher pitches. Once the improper pitcher delivers a pitch he becomes the proper pitcher