= Solution
Take the standard <successor structure> and its disjoint union with one two-sided integer successor chain:
$$
\boxed{\mathcal N,\qquad\mathcal N\ \sqcup\ (\mathbb Z,z\mapsto z+1).}
$$
Both interpret the named zero in the natural-number component. The <classification of models of the successor theory> describes every model as this distinguished one-sided chain together with an arbitrary collection of two-sided chains. Its axioms are <injectivity> of $s$, zero outside its range, a predecessor for every nonzero element, and absence of finite cycles. These axioms are complete; equivalently the theory has the successor quantifier-elimination property explained in the concept proposal.
Consequently both displayed structures are models of the standard <complete theory>. They are not isomorphic: any isomorphism must fix zero and hence every element $s^n(0)$. The first structure has no other elements; the second has the entire extra integer chain. Being in that additional component is an isomorphism invariant even though being outside all standard iterates of zero is not expressible by one <first-order formula>.
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