Any action a character takes is a result of player choice. The player may be attempting to play a consistent character, but it still comes down to what the player chooses to do. These actions can be justified however the player wants, but it's still a choice made by the player. A player can freely say, "It's what my character would do," and they would be entirely correct. It's still a choice by the player. Immersive roleplaying is simply agreeing that "it's what my character would do," is the only acceptable justification for the choices the player makes. It's nothing more than that. It's also no different than agreeing that HP is the only justification for the choices the player makes. They're the same thing, an explanation for why a player chose any particular action for their character. None of these things provide any insight into whether the player has any narrative influence, only that they are involved in the narrative as established by the DM.
So why again are we on this tangent?