I think I got some insight into this from a strange acting class. This class focused on a single exercise, where you simply paid close attention to your scene partner, let yourself be affected by what your scene partner said and did, and then breathed your honest reaction into the words in your next line, regardless of what your words are.

Doing this, you got into the habit of being completely absorbed in each moment, but without trying to make a big deal out of anything. Sometimes you’d have a wild scene with yelling or crying or who-knows-what, and sometimes you’d just talk as normal. Sometimes you might even be bored, and you’d let that boredom show in the way you said your line.

Each moment became meaningful, but there was no pressure. The trick was to not think about extracting any particular result from each moment, but just to pay close attention in each moment to what’s really happening.