Right on the button with that one - you’re right, there’s nothing in the rule book about turning the shoulder, or any other part of the body ONCE THE PITCHER HAS SET. He/she must either deliver, step off, or with distance and direction throw to the bag.
With respect to that last one - throwing to the bag, there are specific guides that umpires use when the first baseman is not in contact with the bag directly … but that’s for another round.
In regards to … "I am leaning toward the latter because turning the shoulders is still called a balk even if a runner is not on first." Correct, to a certain extent. With a runner on third, a southpaw has the same rules applied to him/her, but without the restrictions of the fake throw, and with no runners on, there is no balk applied after a pitcher comes set. In fact with no runners on, a pitcher does not have to come set with a stop motion at all.
These considerations are applied to MLB and any exceptions due to Federation rules or other league play is another matter.