You missed my point. I am not arguing that naturalized means natural born. I agree that if he were naturalized, he would not be eligible to be president.
I was simply responding to an allegation that his mother called in the birth announcement to make it look like he was born in the USA in order to qualify him for welfare. My point was simply that he did not have to be natural born in order to qualify for welfare benefits, and hence his mother didn't have any incentive to try to make it look like he was born in the US, unless she were somehow forseeing the possibility that he might want to run for president.
I have yet to address the welfare eligibility issue. So far I’ve escaped it. But if you want to debate welfare, note that the law was different in 1961.
Under the laws in effect at the time, if he wasn't born in the US, he wasn't a citizen at all. Unless later naturalized, he wouldn't be one today.