Summary: We give definitions that distinguish between two notions of indiscernibility for a set \(\{a_\eta \mid \eta \in{^{\omega>}\omega}\}\) that saw original use in [S. Shelah, Classification theory and the number of non-isomorphic models. 2nd rev. ed. Amsterdam etc.: North-Holland (1990; Zbl 0713.03013)], which we name \(s\)- and str-indiscernibility. Using these definitions and detailed proofs, we prove s- and str-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP\(_{1}\) or TP\(_{2}\) that has not seen explication in the literature. In the appendix, we exposit the proofs of Shelah [loc. cit., App. 2.6, 2.7], expanding on the details.