According to Solovay-Kitaev theorem it is possible to approximate any unitary quantum gate by sequence of gates from small set of another gates. The approximation can be done with an arbitrary accuracy $\epsilon$.
One of such set of gates is composed of Hadamard gate, phase gate ($S$), $\pi/8$ gate ($T$) and CNOT gate. However, it is also true that $S=T^2$ because $T$ gate is a rotation around $z$ axis by $\pi/4$ and $S$ gate a rotation by $\pi/2$ around the same axis.
Since $S$ gate can be composed of two $T$ gates, why do we add $S$ gate to the set? It seems that a set containing only $H$, $T$ and CNOT is equivalent. What am I missing?