“Jindal is a citizen and not a natural born citizen.”
Is he a naturalized citizen? No. Then he must be a natural born citizen, because there is nothing else for him to be. No other category exists.
So why did the founders put two types of citizens in the Presidential Clause?
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.
Actually, there is. Citizen and Natural born citizen. Under citizen one can either be naturalized or born on US soil to one citizen parent and one non-citizen parent and still be a citizen, but not natural born citizen. Both parents must be citizens to qualify as natural born citizen.
There are 14th Amendment citizens according to the Supreme Court, which is naturalization (by law) at birth.
"U.S. Supreme Court
Kawakita v. United States, 343 U.S. 717 (1952)
Kawakita v. United States
At petitioner's trial for treason, it appeared that originally he was a native-born citizen of the United States and also a national of Japan by reason of Japanese parentage While a minor, he took the oath of allegiance to the United States; went to Japan for a visit on an American passport, ..."
And
"First. The important question that lies at the threshold of the case relates to expatriation. Petitioner was born in this country in 1921 of Japanese parents who were citizens of Japan., He was thus a citizen of the United States by birth, Amendment XIV, § 1 and, by reason of Japanese law, a national of Japan. See Hirabayashi v. United States, 320 U. S. 81, 320 U. S. 97.and law. While a minor, he took the oath of allegiance to the United States; went to Japan for a visit on an American passport, and was prevented by the outbreak of war from returning to this country. During the war, he reached his majority in Japan, changed his registration from American to Japanese, showed sympathy with Japan and hostility to the United States,..."
Here we see SCOTUS in the modern usage of "native-born." The Supreme Court further stated that Kawakita was a citizen by virtue of the 14th Amendment. Kawakita had duel allegiances.