Hoover 1,
This might be a great opportunity to pull out the rule book and study it. Quoted below is a portion of the FED rules regarding balks, and the quoted section covers most of the concepts necessary to answer your question. In elaboration:
1. Once the pitcher steps off legally (I'm gonna assume here that the pitcher did the "up and over move" correctly), he can throw anywhere because has temporarily become a fielder.
But if he is in contact.....
2. First base is special only because the pitcher may not feint a throw to first. He can feint to second or third.
3. He can't throw to an unoccupied base except to put out or drive back a runner (the drive back part only is true in FED and not in OBR). There is nothing special about first base in this regard.
So in your situation, it isn't a balk in more than one way. First the pitcher was clearly off the rubber before making any throw, so throwing to any base is legal. Even if he were still in contact, throwing to third would be OK as long as he is trying to put out or drive back the runner. Practically speaking, that means that unless the runner is camped at second with no lead, a throw to third is usually legal.
About the only way the pitcher may have balked is to have not stepped off legally.
From NFHS Rule 6-2-4:
ART. 4 Balk. If there is a runner or runners, any of the following acts by a pitcher while he is touching the pitcher’s plate is a balk:
a. any feinting toward the batter or first base, or any dropping of the ball(even though accidental) and the ball does not cross a foul line (6-1-4);
b. failing to step with the non-pivot foot directly toward a base (occupied or unoccupied) when throwing or feinting there in an attempt to put out, or drive back a runner; or throwing or feinting to any unoccupied base when it is not an attempt to put out or drive back a runner;