"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside."

Correct. They are citizens. Obama was a citizen at birth. Hence a natural-born citizen

Not all "citizens" are "natural-born citizens."

Also correct. Citizens who are citizens at birth are natural born citizens. Citizens who have to go through a naturalization process (such as immigration) are not natural born citizens.

What you are trying to argue is that they are only the same thing, which is not true

I agree, they are not the same thing

All natural born citizens are citizens. Not all citizens are natural born citizens

"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside."

Correct. They are citizens. Obama was a citizen at birth. Hence a natural-born citizen

Not all "citizens" are "natural-born citizens."

Also correct. Citizens who are citizens at birth are natural born citizens. Citizens who have to go through a naturalization process (such as immigration) are not natural born citizens.

What you are trying to argue is that they are only the same thing, which is not true

I agree, they are not the same thing