The language before about people born in the United Staes deals with citizenship in the United States. The courts have drawn a distinction between the rights of citizens and the rights of persons. The Rights of persons apply to all people including non-citizens. So, while it is clear that citizenship only applies to those who have been born it is not clear that personhood is limited to those who have been born.
Interestingly, the courts have upheld the property rights of the unborn. They can inherit and have standing in probate.