I am sure opinions will vary on this.
I use the name of the character in all the cutscenes, but I use 'player' in all the scripts. That way when I am testing a mission I can play as a temporary unit that might have more/different equipment than the player character and the scripts will still work because this temporary unit is now the player. The one gap in my knowledge on this is Muti-Player. I have written no scripts for MP missions so I cannot say how player is treated there.