Nobody is debating the definition of running lane. We are simply debating the umpires judgment:
(11) In running the last half of the distance from home base to
first base, while the ball is being fielded to first base, he
runs outside (to the right of ) the three-foot line, or inside
(to the left of ) the foul line, and in the umpire’s judgment
in so doing interferes with the fielder taking the throw at
first base, in which case the ball is dead; except that he
may run outside (to the right of) the three-foot line or
inside (to the left of) the foul line to avoid a fielder
attempting to field a batted ball;
Turner reached first at the exact moment the ball was caught. In fact, the first baseman's glove swept into Turner (who was on the base at the time) as a result of receiving a poor throw from 3rd.
It's simply a matter of poor judgment.
By the way, my comment on carrying the bat to first was in jest. Even though it appears it's OK while the runner, in the act of stepping on first, is out because of he made contact with the glove--which happens ALL THE TIME with errant throws to first from all over the infield. Do you think anybody is paying attention to the running lane when the SS makes a bad throw?