= Solution
Apply downward Löwenheim-Skolem to the extension from part (i), keeping all standard <integers> and the new prime $c$. The <first-order language> and this prescribed <subset> are <countable>, so there is a <countable> <elementary substructure> $N$ containing them. In particular $N\equiv\mathcal Z$ and contains an element outside the standard copy of $\mathbb Z$.
No <ring isomorphism> $N\to\mathcal Z$ can exist. Such an <isomorphism> fixes $0$ and $1$, hence every nonnegative <integer> numeral. It also fixes each negative <integer>, which is the unique solution of $x+\bar n=0$ for the corresponding positive $n$. Consequently it fixes the whole standard copy. The image of $c$ would be some standard <integer> $a$, already the image of that same named standard element, violating <injectivity>.
Thus $N$ and $\mathcal Z$ are two nonisomorphic countably infinite <first-order models> of $T$, so
$$
\boxed{T\text{ is not }\aleph_0\text{-categorical}.}
$$
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