The fact that Naturalization law adopted in 1795 removed the phrase “natural born citizen” tells us that this phrase did not have the same meaning as the phrase “born a citizen”. Otherwise, why change the law after only five years?
Few years earlier, Hamilton proposed that the eligibility requirement for US Presidency includes the phrase “born a citizen”. Instead, John Jay’s proposal was adopted. In his letter to George Washington Jay proposed that eligibilty be given to “natural born Citizens”.
Eligibility requirement is a national security measure, a strong check preventing a person with dual allegiance from taking command of US military.
Neither Obama nor Cruz are eligible for US presidency.
The law was about Naturalization. It was called a Naturalization Act. They could have removed the comment about Natural Born Citizen because it wasn't relevant to a Naturalization Act.