I had a really good scene recently, set in a TK Max, where I was really really in the character. And that's what made the scene good to me. If I think about my sort of "golden age" of improv, when I was feeling the most proficient, it was when I was acting really well. It's not me the player thinking, "I should be sad now" and then being sad. It's the character reacting to what just happened. There's a huge difference, and I think I forget that sometimes. I think when I talk about looking for presence, that's what I mean. It's not just showing up, it's inhabiting the world. So maybe my focus on presence is misguided, there's an activeness to it, that I don't have when I just try to be "present" in a meditative way. I'm always in me, the player's body, not the character. So I think what I really mean is "act". Acting in character makes you incredibly good at making up things that sound believable, and reacting in a way that is believable. There's no longer a translation layer of me the player thinking of what to do next, which is incredibly freeing as well. In that relaxed state of knowing that you will react naturally, you are free to listen better.
What I'm saying is, if you're an improviser then take an acting class. Then try to use it.