Rules of the game (MLB)

I thought it would be fitting for my first post to discuss a rules question I've thought about for baseball for quite a long time. Here's the situation: Runner on first, less than 2 outs.  The pitcher accidentally goes into the windup, and the runner takes off for second.  (Or the runner takes off, then the pitcher goes into the windup.)  There is a strange combination of blazing speed in the runner and very slow motion in the pitcher's windup.  The runner reaches second base before the pitch reaches the plate.  The batter hits a lazy fly to the outfield.

Now: does the runner need to tag up at first base or at second base?  The real issue here involves when the runner gains possession of a base and when a "play" in baseball starts.  It could be that as soon as the pitcher begins his motion, the situation is "frozen" and anything that happens after that requires the runner to tag up at first.  Or, it could be that there are no "plays" in baseball, that the action is continuous and that the runner can claim any base he can reach before the ball is struck by the hitter. The second option raises interesting implications for the situation when there is a runner on third with 2 outs.  If he steals home and crosses the plate before the hitter strikes out, does the run count?  That situation is bound to have occurred during some game somewhere, and it should give clues about what the verdict should be in the case with the runner on first with less than 2 outs.