As noted, Trumbull and Bingham had different views. And the amendment was passed by several hundred members, some of whom sided with Trumbull and Bingham, and some perhaps sided with Howard. Thus the intent of the writers of the amendment from what a few of them said gets only so far. The job of determining the meaning of legislation and the Constitution is that of the US Supreme Court.<quoted text>While the Citizenship Clause was intended to define as citizens exactly those so defined in the Civil Rights Act, which had been debated and passed in the same session of Congress only several months earlier, the clause's author, Senator Jacob M. Howard of Michigan, phrased it a little differently. In particular, the two exceptions to citizenship by birth for everyone born in the United States mentioned in the Act, namely, that they had to be "not subject to any foreign power" and not "Indians not taxed," were combined into a single qualification, that they be "subject to the jurisdiction" of the United States, and while Howard and others, such as Senate Judiciary Committee Chairman Lyman Trumbull of Illinois, the author of the Civil Rights Act, believed that the formulations were equivalent, others, such as Senator James R. Doolittle from Wisconsin, disagreed, and pushed for an alternative wording.
The US Supreme Court determined in the Wong Kim Ark case (which was by a vote of six to two, one justice not voting), that the meaning of Natural Born comes from the common law and refers to the place of birth. It also ruled that the meaning of "subject to the jurisdiction" did not exclude the children of aliens born in the USA. Unless they were the children of foreign diplomats, they also were subject to the jurisdiction. That is the law, and the chance of the US Supreme Court reversing it is so low as to be considered laughable. But, unless and until the US Supreme Court reverses that ruling (or we pass a Constitutional amendment to overturn it, which is equally unlikely), that will continue to be the law.