“A natural born citizen is a person who became a U.S. citizen at birth and did not need to go through a naturalization proceeding later in life.”
Here’s the issue: Both 14A §1 Birthright Citizens and Art. II, §1, Cl. 5 Natural-born Citizens are persons who became U.S. citizens at birth and do not need to go through a naturalization proceeding later in life.
A separate legal effect: While 14A §1 Birthright Citizens are subject to foreign sovereignties at birth, natural-born Citizens are not, being born exclusively from Citizen-parents.
The 14th Amendment doesn’t grant citizenship. Either citizenship is natural or naturalization laws (USC 8 for us today) come into effect.