O'Brady -- you make some interesting points. I enjoy the dialogue.
Couple of things: Please re-read. I didn't state that "a play has to be made."
And in the scenario you describe with a traditional steal and no throw -- yes, that would be a steal. (Although I've not seen "traditional" and "non-traditional" steals mentioned in the rulebook.)
Bottom line, to me the posted scenario is simply a batter trying to stretch a single into a double -- and succeeding. The fact is that the ball was still live and the runner was still in motion -- so he had never stopped.
I also think the rule you quoted disqualfies this from being a stolen base ... because his advancing was indeed "aided by a hit". Note, it doesn't say "caused" it says "aided". And that's pretty much what happened here.
So for me, score it as a single that got stretched into a double.
Any other views out there?