... we are talking about the law, and often in law terms are used that do not have the same meaning as we give them in normal conversation. Example: A "person" includes humans and corporations in the law, but we all know that a corporation is not a living, breathing person, as we would say in common (non-legal) discussions.
Therefore, you cannot pull the definition of a term out of a common dictionary and assume it has the same meaning as it does in a specific legal issue.
Naturalized includes BOTH acts of Congress that grant citizenship to foreign-born adults who go through a formal naturalization process, AND acts of Congress that uniformly grant citizenship to classes of people who do not have citizenship by birthright (such as those born not on US soil to US military servicemen).
Look up Title 8 and check out the nationalization statutes (around section 1400 or so).
Regarding your example of Bush being born in Mexico to US parents , this is the EXACT situation of George Romney. Do you not READ?
Bush would not be considered a foreigner because he would have been granted US citizenship at birth by an act of Congress (i.e. naturalization) because he would have belonged to a certain class of people (those born to US parents on foreign soil). So no, he would not have been a foreigner. Yes, he would have been a US citizen (through naturalization). No, he would not have been a natural born citizen and not been eligible to be president. This is probably why George Romney dropped out of the Republican Party nomination process in 1968, even though he was doing well.
The founders were born British subjects, as were their parents (unless a subject or citizen of another country) because the United States did not exist at the time of their birth. For that reason, they granted an exemption to anyone living at the time of the adoption of the Constitution, since nobody would have been eligible otherwise.