I want to clear up your confusion, if there is any. Obviously citizenship by birth is not the same as the circumstances of birth, and I never made any attempt to make them equivalent.
In fact you might combine them to say “citizenship by the circumstances of birth”.
Does that help?
I should have given my core question, namely, does “citizenship by birth” mean born in America? Where’s the proof of that? A lot hangs on the meaning of that term.