Not at all. But "natural born citizen" being the same as "citizen at birth" still differs from the more generic "citizen" in that naturalized citizens qualify for the latter but not the former.
Also, it is somewhat off track of a good debate to have one person telling another that something is not personally relevant to that other person. Reminds me of thought police.
Everyone is entitled to their own opinion, and I do not dispute that. But an opinion is irrelevant compared to fact in the sense that it cannot change the fact - I could have the opinion that no one should be allowed to be President who hasn't served in the military, but that opinion has absolutely no bearing on anyone's legal eligibility to be President. Similarly, people who are imposing an extra-legal and extra-Constitutional test to the Presidency are welcome to do so for themselves, but that doesn't affect the actual eligibility requirements.
My brother and I who were born in the USA and were in military service in WWII were always considered plain ‘citizens’ or citizens at birth using your term. Our parents were not naturalized immigrants. However, contrary to your statement we were not the same as/identical to a ‘natural born’ citizen. I can agree that a ‘natural born’ citizen is a ‘citizen’ when born but not that a ‘naturalized’ citizen is a ‘natural born’ citizen. I recall an old adage that goes something like ‘facts are facts just don’t confuse me with your opinions’ and it seems we are at this point in our discussion.