“§ 1473. It is indispensable, too, that the president should be a natural born citizen of the United States; or a citizen at the adoption of the constitution, and for fourteen years before his election. This permission of a naturalized citizen to become president is an exception from the great fundamental policy of all governments, to exclude foreign influence from their executive councils and duties. It was doubtless introduced (for it has now become by lapse of time merely nominal, and will soon become wholly extinct) out of respect to those distinguished revolutionary patriots, who were born in a foreign land, and yet had entitled themselves to high honors in their adopted country.”
Which means that if Obama was born in Kenya, then he isn’t a natural-born citizen because federal law at the time only gave U.S. citizenship at birth to a child born abroad with one U.S. citizen parent if such parent had lived in the U.S. for at least 5 years after the age of 14 (and Obama’s mother hadn’t yet turned 19 years of age when he was born). However, if Obama was born in Hawaii, then he’s a U.S. citizen at birth and thus a natural-born citizen.
“... thus, a natural-born citizen is someone that is a citizen at birth, while a naturalized citizen becomes a citizen later in life.” By mistating your premise to include that which you wish to establish as singular truth, you have deceived readers and yourself. Naturalized is a process reached through statute, ie, some legal statute establishes citizenship. By statute is not a state achieved by meeting the primary definition of natural born. Even you can think up a case where someone born on American soil but not a natural born citizen at birth and not a citizen at birth can become a citizen by statute passed later in life yet reaching back to date of birth, state of parents’ citizenship, and location. [HINT: think children born to diplomats and years later Congress establishes a statute covering siad child born to a diplomat of a foreign country.]