I am split down the middle (with these defintions) as a Power Gamer / Character Actor.
I play games for the moment. I don't necessarily look at big picture - although after being a DM for 30+ years I tend to do it without thinking. But my purpose for play is immersion - be the character, and live the fantasy world through their eyes, fell what they feel. In one case (in solo play) I played for 2 months without a single combat, just living the life, and making the choices of the character.
I also like a good combat, and like to have combat capable characters. And unlike others combat doesn't take me out of that moment to moment immersion. I tend to make all tactical choices within character, and I tend to play combat type characters (Monk who think tactically, Wizards with a smattering of battlefield knowledge) so making game mechnical tactical decisions is a stand in for the character making similar choices.
I go Storyteller out of playtime- thinking of possible scenes or character arcs for the character and discussing them with the DM to work into his campaign. But once at the table, totally in the moment.