= Solution
Enumerate the standard positive primes as $(p_n)_{n<\omega}$. For each <subset> $A\subseteq\omega$ consider the <partial type>
$$
q_A(x)=\{x\ne0\}\cup\{\bar p_n\mid x:n\in A\}\cup
\{\neg(\bar p_n\mid x):n\notin A\},
$$
where <divisibility> is the <ring> <logical formula> $\exists y\ (\bar p_ny=x)$.
Every finite <subset> is realized in $\mathbb Z$: take the product of the finitely many primes required to divide $x$, or $1$ if there are none. This is nonzero and is not divisible by any of the distinct finitely many primes required not to divide it. <Compactness theorem> supplies a <complete type> extending $q_A$.
Distinct <subsets> $A,B$ yield distinct complete extensions, because at an index in their symmetric difference they require opposite <divisibility> statements. There are $2^{\aleph_0}$ such <subsets>, so there are at least that many <complete one-types>. There are at most that many because the <first-order language> has only countably many <logical formulas> and a <partial type> is a <set> of them. Hence
$$
\boxed{|S_1(T)|=2^{\aleph_0}.}
$$
In particular the number of <complete one-types> is uncountable, despite every <partial type> realized in the particular <countable> <first-order model> $\mathcal Z$ being a <principal complete type>.
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