It is YOU who is mistaking the formal naturalization process for for all forms of citizenship granted through statute enacted by Congress under the plenary powers granted to it by the naturalization clause of the US Constitution.
In fact, ALL forms of citizenship other than “natural born” citizenship exist through the operation of federal statute law, and are therefore properly thought of as forms of naturalized citizenship. This includes anyone granted US citizenship although born in a foreign land (other than the children of diplomats). This is true whether the child has one or both US citizen parents. Absent this statute, no child born abroad (other than the child of diplomats) would be a US citizen at all.
To repeat: ALL forms of citizenship that derive from the operation of statute law are a forms of NATURALIZED citizenship.
Perhaps not quite all. The 14th amendment declares that all persons born in the US and subject to their jurisdiction are citizens. That category is likely not quite congruent with "natural born citizens".