In my opinion, even if Obama was born in Kenya and had an officially certified Kenyan birth certificate in his hand, he would still have been legally allowed to be the President of the USA for his two terms because Obama’s mom was an American citizen, on the same legal basis as Cruz, who was born in Canada.
Remember when thinking about the issue, naturalization does what nature didn’t.
No naturalization + undoubted US citizenship -> natural-born.
Foreign born are naturalized, period.