Now imagine this scenario... a RHP runner on 1st. As RHP lifts left leg, runner breaks to 2nd on 1st move. RHP spins and throws to 2nd ... not a balk provided his left foot did not completely cross the back edge of the rubber. Have yet to see that in an actual game, but have played it over and over in my mind - and can't see where there is a rules infraction.
Agree with what you said regarding the original situation. Would have to see it, but I think maybe a balk if he hangs the leg toward 2nd, but keeps leg up and then just shifts toward 3rd base and throws there. Umpire lore says it's hard to balk to an occupied 2nd base, but this might be an exception.
He's fine if he steps directly toward 2nd base. He doesn't even have to make a "fake throw"; the act of stepping toward 2nd base is considered to be the "feint." Once he has made that step toward 2nd, he is legally disengaged and can throw wherever he wants.
By the book, he could also just lift his leg, step straight to third, and throw there ahead of the advancing runner. But - this is likely to be balked in a lot of places, even though technically legal. If considering doing it, it might be a good idea to have a chat about it with the base umpire beforehand.
Regarding the first-move steal from 1st base: the non-pivot foot crossing the rubber is irrelevant if making a play at 2nd base. Pitcher just has to step in the direction of 2nd base and throw ahead of the runner. Same caveat about maybe getting balked, even if making this move legally.