Thank you for bringing this up. The difference is so subtle!
Where I draw the line is a user’s intention to be in the company or enjoy solitude. This idea is coming from the “peace of mind” litany that we, UX Designers, have to keep in mind at all times. The user wants some peace of mind. The user wants to be left alone without interruptions.
For the user who wants to be alone, to deal with their possessions, businesses, processes alone, without witnesses, it would be “My drive”. For the user who needs an assistance and as such will have to tolerate a presence of a second entity: “What would you like to have today?”.
Coming from this angle, I would like to see “My wishlist” and “My lists” on Amazon, and “My music” in iTunes (could it be that the latter being “Your music” is a gentle semantic reminder that the music rights belong to someone else?).