What is a local inference system?
$\mathsf{Steps}(\Pi)$ is defined just above, in the paragraph below Definition 4.1. It's the set of node labels in the proof, i.e. the set of propositions that are proved as lemmas in that proof.

Thus a local inference system is a set of propositions such that every deduction $T \vdash u$ can be proved from lemmas that only involve terms that are subterms of the proposition $u$ or of the premises $T$.