Is there a way to have audition start recording when

No there isn't, I'm afraid. To achieve a realistic result with this would involve what happens on hardware recorders - a rolling buffer. The idea is that all the time, whatever would be recorded when you start actually comes from the buffer. You need this to take account of the finite time an actual start takes - you have to detect the sound first, and then start the recording. What happens with the buffer is that the previous couple of seconds of everything (even you not playing) is always in it, so it can call on that to get the 'correct' starting point. So an actual note triggers recording proper, and what's before it comes from the buffer. That's why, with hardware recorders, you get a flashing red 'record ready' light - it means that the buffer's getting filled.

There is no other way of doing this. If there was, then the hardware manufacturers would have come up with it ages ago. And as a bit of code, whilst it's obviously possible to implement something like that, there'd have to be quite a demand for it before it got seriously considered.