Balk?

[Gre144]

Fed rule only

Situation 1

B3 on third heads for home. Pitcher who is on mound steps towards home and throws to cathcer without starting any preliminary motions and without coming to a complete stop. B3 is thrown out at home.

Is this a balk(and if so please tell me which Fed rule to look up)? I feel that it should not be a balk since the pitcher has a right to step towards a bag(in this case home) while a runner is advancing to that bag

Situation 2- Assume situation 1. Assume, if we can, that situation 1 is not a balk. The batter hits the ball from the throw of the pitcher who is trying to retire R3 going home. How do we deal with this situation since F1 is not technically pitching the ball but is rather throwing the ball to retire B3 coming home?

Thanks,

Greg