I believe it to be true; according to Blackstone [here]:
As to the qualifications of members to sit at this board: any natural born subject of England is capable of being a member of the privy council; taking the proper oaths for security of the government, and the test for security of the church. But, in order to prevent any persons under foreign attachments from insinuating themselves into this important trust, as happened in the reign of king William in many instances, it is enacted by the act of settlement,l that no person born out of the dominions of the crown of England, unless born of English parents, even though naturalized by parliament, shall be capable of being of the privy council.Clearly the state of
natural bornis not a product of naturalization; to claim that it is is to claim that the Congress can alter the Constitution by normal legislative acts (i.e. without amendment) which would defeat the purposes both of having a written constitution and of defining some class of citizen of which naturalization ruled over.
Which is why I also doubt the eligibility of both Obama and Cruz.
When in doubt, taking the most stringent definition is usually the best option: everything that fits within the strict-subset by nature must fit within the containing super-set, but the reverse is certainly not true.