We give necessary and sufficient conditions for the Terwilliger algebra of a quasi-thin Schurian association scheme to coincide with: (a) the centralizer algebra of a point stabilizer of its automorphism group, and (b) its subspace $T^0$. As a by-product, a full characterization of triply-transitive quasi-thin association schemes is given. Using the latter, we provide new infinite families of triply-transitive association schemes.