To me, ‘my’ Doctor is a term for the Doctor that I’m most invested in emotionally, that I feel closest to (I guess), whereas ‘the’ Doctor is the one I think of whenever someone mentions him/her to me — basically the definitive incarnation. I’m not sure how others would describe the ‘my Doctor’ term, but that’s what it means to me.
I’ve only just started watching a Classic Who, and am waiting for the opportunity to get more DVDs, so it might change later on. I’ve read a lot about it, though.