Balk or not

[clevbrown]

NFHS rules. Runner on 2nd, pitcher pitching from the stretch. After going to the set position, the pitcher pivots on the rubber, steps toward 2nd base and fakes a throw to 2nd. The runner takes off to 3rd not realizing the pitcher was faking a pick off move to 2nd. The pitcher, no longer in contact with the rubber throws the ball to 3rd and the runner is tagged out. Is this a balk or a legal play?

My opinion:
NFHS Rule 6.2.4.b says it is a balk if there is a runner on and the pitcher throws or feints to an unoccupied base when it is not an attempt to put out or drive back a runner. So, in my opinion it is a balk because he stepped toward 2nd, then threw to 3rd. Had he stepped to 3rd and thrown to 3rd, he would not be a balk because he was making an attempt to put out a runner.

I have heard 2 arguments from other umpires on this.
Argument 1: it is a balk because the pitcher must step behind the rubber before he can throw to an unoccupied base.
Argument 2: it is not a balk because the pitcher while faking a throw to first, stepped toward second and after his pivot foot left the rubber during the step toward 2nd, he is now a fielder and can throw wherever he wants to.

Thoughts?