When I was recording a level, I quitted and the game asked me whether to save the unfinished run. I didn't want to save it, so I clicked 'No'. When I activate the recording system again, the game didn't prompt me on whether to save an unfinished run when I exited the level OR the game didn't save a completed run.
When at least recording online, I didn't like it when the recording system didn't save a run after I restart several times within a level (without exiting the level).
Since the Awesome Time/Score is secret, you will have to beat it in order to find out what it is. Just practice every day, and be ready to keep pressing the 'Restart' button...A LOT!