I think that people who are not practicing musicians themselves tend to respond to how a musician plays a piece rather than stuff about the piece itself. Being a musician who wears many hats, I bounce back and forth. If someone plays a piece that I wrote, I tend to ignore the notes and rhythms of the music and focus on the interpretation and the playing. Still, I get a special thrill when someone finds something in a piece that I didn't even know was there, and then my attention is temporarily re-directed to the music itself. When I am listening to a performance of a piece I know very well by another composer, and there is something novel about the interpretation, my attention also goes right to the piece. When I'm hearing something I have never heard before, I almost always focus on the piece, and let the playing take the back seat.
When I play a solo concert I think mostly about shifting, intonation, sound, bow distribution, phrasing, ensemble, and I think about expressing myself, relegating the composer to the role of the person who provided the notes and rhythms, and did all the work to have them fall in the right places. When I play an orchestral concert I tend to pay more attention to the music than to my playing. It's all very confusing.
All in all I think that it is the performer's art that is the most important. I don't think that I'm alone among composers (living or dead) when I imagine how someone might play a phrase while I am writing it. The notes and rhythms are essential, but the musical gesture is what really matters. Someone can design the most beautiful dance costume, but if it is worn by a person who can't dance, it means nothing. Then again, if it is worn by a wonderful dancer, the costume becomes part of the whole experience of the dance.