Unfortunately, nonstandard models will survive any such attempt. This is guaranteed by the Löwenheim-Skolem Theorem which says that if a countable first-order theory T has an infinite model then it has one of every infinite cardinality. Since an uncountable model necessarily has nonstandard elements, this guarantees that there is a nonstandard model of T (and even countable ones).
Actually, in your case you need a "two-cardinal" version of Löwenheim-Skolem. In your ZFC example, the natural numbers form you move to a theory which interprets arithmetic inside a definable substructure of (the set ω). The definable substructure of such a model which might still be countable even if the model itself is uncountable. Nevertheless, one can still blow up the size of the natural number substructure via the ultrapower construction, for example.
To evade the Löwenheim-Skolem Theorem, one has to move beyond first-order logic. For example, in infinitary logic one allows infinite disjunctions such as $$\forall x(x = 0 \lor x = S0 \lor x = SS0 \lor \cdots)$$ which ensures that the model is standard. Also, second-order allows quantification over arbitrary sets under the standard interpretation, which again prohibits non-standard models. (See this related question.) This is the characterization of N most commonly used by working mathematicians.