What you are saying is true (as you allude to with first-order logic) in any context in which Godel's completeness (not incompleteness) theorem holds. Not all logics have versions of Godel's completeness theorem (e.g. second order logic with full semantics). You can argue that philosophically in systems where Godel's completeness theorem fails that the article's statement is valid. But yes that's why it's only true in "certain senses."

More generally the philosophical question of truth revolves around whether there is a single, true set of standard natural numbers that corresponds to reality, and therefore all nonstandard natural numbers are "artificial" in some sense, or whether even standard natural numbers exist only in a relative sense.