The founders had an ambition. Limit the presidency to people with no natural inclination NOT to remain fully loyal to the ideals of the republic AND to the Constitution. That means the president should have no vestige of loyalty to any foreign government, people, or land, through foreign birth, foreign parentage, or long residence in a foreign land. NBC accomplishes that ONLY if the NBC is born on US soil of US citizen mother and US citizen father.
That requirement is a level above a mere citizen born on US soil of foreign parents, or born abroad of US citizen parents, and especially born abroad of one foreign and one citizen parent, or born with or acquiring dual US and foreign citizenship.
Ted Cruz does not qualify to become president and should not be allowed to run for the office even though votes of the people only determine who can become delegates in the presidential election. Cruz experienced birth in Canada of a US Citizen mother and a Canada citizen Cuban-native father. He is not a NBC. Neither is Rubio, so he never could qualify for the presidency because he experienced birth in the USA of Cuban citizen parents.
Our problem: Congress has not made a law obligating an election qualification commission to ensure that all candidates qualify for office prior to having the right to run for election.
Congress does not have the authority to define Natural Born Citizen in any way other than the above way, in keeping with the widely respected Emmerich Vattel's translation and modernization of the Law of Nations.