When is courtesy runner legally in game?

[drh898]

Situation came up in yesterday's game, catcher singled, once play stopped coach called time and requested courtesy runner for the catcher. Number 8 came out and replaced catcher at first. PU (I'm the BU) announced to scorer that #8 is courtesy runner for the catcher. Just before the first pitch to the next batter coach calls time and announces he wants #5 to be the courtesy runner instead of #8. BU allows this. Between innings the other coach questions the replacement of the CR and PU now claims that both
#5 and #8 have entered the game and were not CRs. (He never called me over to confer and we are told by our association that the BU never joins a discussion with coaches unless asked). So now there is confusion as to who has entered the game, etc.

I believe that once a CR replaces a pitcher or catcher that CR cannot be then run for in that sequence unless injured. Correct me if I'm wrong there. In discussing this with another veteran umpire he said that the coach could replace the CR before the first pitch to the next batter. So, in a nutshell, when is a player legally in the game? When she is announced by the PU to the scorer or after the first pitch to the next batter?