I realized this while I was reading Being and Mindfulness, by Judith Warner. She says:
It has dawned on me lately, meditating on the Metro, thoughts silenced so completely that I can hear every page being turned by passengers up and down the car (I am above reading — I am present to myself) that being fully in the moment, all senses turned on, feeling your hands in your lap and the ground under your feet, is a very good way of not being there at all.
That there was my mistake also: in my rush to be present, I ran off after "presence", after "being present" and "practicing mindfulness". In that rush, I lost what I was actually running after (and ended up taking myself too seriously).
Maybe I need to meditate more.