This sentence doesn't make sense. She was never naturalized (and her father's US citizenship status is immaterial) because she was born on US soil. She acquired US citizenship through jus soli, or so was held in Ark.
As for native born vs. natural-born, Elk uses these two statements interchangeably and in the dicta of the decision, the majority opinion makes mention of the fact that they're interchangeable phrases. I would cut & paste that particular line, but I'm on a iPhone and it's just too darn hard. But, if you read the entire decision, it's in there.
A naturalized person is an immigrant who comes to America and goes thru the process to become a citizen, as her father did. He completed the process one year before her birth. Because her mother never became a US citizen, but her father did, and she was born in the US, she became a naturalized citizen by the merits of her father. Had her mother also become naturalized, she would be Natural Born, as a citizen born on US Soil to two citizens (Jus Soli).
Ark was never granted Natural Born Status, rather naturalized citizenship.
The fact that Elk interchanged the two did not translate to US law. The distinction is clear. Nobody denies that she would have citizenship status.
Obama’s father was not a US citizen, ever. His mother was not back in the US long enough to confer Naturalized status to Barak Obama II, therefore, he is not Natural Born, and I believe he is not a US Citizen at all. This is just my opinion. He could produce evidence that shows something unexpected. But from what we know, I think he is an Illegal Alien.