It's not even true then.
I am a US Citizen. I live and have always lived in the United States. My wife is Canadian. She is still residing in Canada and I visit across the border spending nearly half my time there until we can get custody issues with her daughter from a previous marriage resolved.
Our son was born in Canada. He is still a natural US citizen at birth, because I am a US citizen, and have lived as an adult in the US for far more than the required number of years. We do have to jump through some hoops to claim that Citizenship and pay a sizable fee for the paperwork, but he isn't granted citizenship by that process. He is already a US citizen, the government just hasn't properly recorded that fact yet.
A Naturalized Citizen is someone who was not born a US Citizen, but becomes one at a later date.
Sorry to burst your bubble, but natural born citizens don’t have to jump through hoops or fill out ANY paperwork to prove their citizenship.
That is the statute law, but the Supreme Court disaggrees insofar as statutory citizens at birth are also considered "naturalized" for Constitutional purposes.
Persons such as your son, while Citizens at Birth,under 8 USC 1401, are considered naturalized at birth. They are citizens through an act of Congress, and since Congress only has power over naturalization, not citizenship in general, they must be so considered. That would even be true if both parents are citizens, a situation where there is no residency requirement for the parents. Such foreign born, citizens at birth, have less protections against loss of citizenhip under the 14th amendment than persons naturalized in the US, because they do not match either of it's terms, born or naturalized in the US.
See Rogers v. Bellei, 401 U.S. 815 (1971).
So your son is not a Natural Born Citizen, and cannot be President. But then again, that's the Good News. :)