= Solution
The distinguished chain generated by zero is <countable>, and every other component in the <classification of models of the successor theory> is a <countable> copy of the integer successor chain. A model of <cardinality> $\aleph_1$ therefore has exactly $\aleph_1$ additional components: a finite or <countable> collection would be <countable>, while the union of $\kappa$ <countable> components has size $\max(\kappa,\aleph_0)$.
Given two such models, match their zero chains in the forced way and choose a <bijection> between their additional components. On each component choose an origin and match corresponding integer iterates; this commutes with successor. The union of these maps is an <isomorphism of structures>. Existence is given by a disjoint union with $\aleph_1$ integer chains. Hence
$$
\boxed{T_S\text{ is }\aleph_1\text{-categorical}.}
$$
The same argument works at every <uncountable> <cardinal>. <Countable> models can still differ by the number of extra components, as part (ii) shows.
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