[K-oneTexas]

Article II Section I Clause V states:

No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.

Obama was born on August 4th 1961 and Hawaii became a state on August 21st 1959. Therefore anyone born in Hawaii after August 21st 1959 is a natural born citizen. That is what the President is claiming. Noting to do with native born.
I don't remember anyone arguing "native born" in the Presidents case. Obama is definitely not claiming 'naturalized'. So I guess I am missing the point of your two cites on the question before us.

[meadsjn]

Therefore anyone born in Hawaii after August 21st 1959 is a natural born citizen.

Not true. A true statement would be: "... anyone born in Hawaii after August 21st 1959, and born to parents (plural) who are citizens of the United States, is a natural born citizen."

Someone who has divided loyalties, or partial allegiance to any other nation (such as one would have if one parent is a British citizen), is not a "natural born citizen", although they might be a "native born citizen".

[El Gato]

Therefore anyone born in Hawaii after August 21st 1959 is a natural born citizen. That is what the President is claiming. Noting to do with native born.

If you have a source where he, or "his people" claim he is natural born, post a link to it, or otherwise provide the source.

AFAIK, B.H. Obama Jr. and his people claim he is native born, not natural born. He might be,but they also claim his father was a subject of the UK, and became a citizen of Kenya after that country gained its independence. Thus BHO Jr, even if native born(born in the US), is not natural born(born in the US of parents who were its citizens at the time)