Stepping off before throwing can have its own problems. If the ball is thrown out of play, the runners advance two bases. Once he disengages the pitcher’s plate, he is considered a fielder. If, however, the pitcher had just thrown from the rubber to first base, it’s only a single base award if he tosses it out of play.
If there is little or no room in foul territory behind first base, I would think twice before stepping off to throw. Step off and hold the ball. The runner will still get dirty by diving back to the base. Maintain contact with the rubber for throws over to first if you can make that play.
It is a balk when… d) the pitcher, while touching his plate, throws, or feints a throw to an unoccupied base, except for the purpose of making a play;
As to the move to second base, he can’t throw to an unoccupied base, but if, as a RHP, he lifts and rotates clockwise toward second he may throw to make a play on the runner, but at the same time, he is not obliged to throw.
Rule 8.05© requires the pitcher, while touching his plate, to step directly toward a base before throwing to that base. If a pitcher turns or spins off of his free foot without actually stepping or if he turns his body and throws before stepping, it is a balk. A pitcher is to step directly toward a base before throwing to that base but does not require him to throw (except to first base only) because he steps.
He can hold the ball and check the runner’s position. After the spin, he can charge the runner and throw to any base necessary to make a play on the runner caught between 1B and 2B.