English common law doesn't say that, and since many were lawyers then they would know that to be the case. If they had meant for the meaning to be outside of that then I would expect that they would have mentioned it somewhere in their writings. If they did, then I'm not aware of it.
You are also aware that your definition would exclude McCain, Jindal, and numerous others.
Yes, Jindal would be unqualified under that definition.
I think McCain was OK in the sense that his parents were both citizens and their home of record was within the US, even though on temporary duty in Panama at the time. It’s no different than if two US citizens had a child while visiting a foreign country. i.e., I think the important thing is two citizen parents, not location. If Zero’s father had been a US citizen this wouldn’t be an issue, obviously.