You're confusing Lie groups with Lie algebras; they're closely related but not the same thing. For example, the $S_i$ are generators of a Lie algebra, and therefore elements of a Lie algebra, but they aren't generators or elements of a Lie group. The relationship is that the algebra expresses the composition of group elements that are infinitesimally close to the identity. You get the group (or at least the part connected to the identity) back by exponentiating the algebra.
The first clue that the $S_i$ aren't group generators or group elements is that in the commutator they are being multiplied together, then one of the products is being multiplied by -1, and then the two products are being added. Groups have only one operation: composition of group elements (often called "multiplication", especially when the group can be represented as matrices). By contrast, algebras have three operations: vector addition (which produces a vector), multiplication of a vector by a scalar (which produces a vector), and vector multiplication (which produces a vector). They are basically vector spaces with a vector multiplication that obeys the distributive law.
In this case the $S_i$ are the basis vectors of the Lie algebra, even though they happen to be matrices. They belong to the Lie algebra su(2) that they generate. They can be exponentiated to form elements of the Lie group SU(2).
The generators are not unique. They are just an arbitrary basis for the vector space of the algebra. You can use other linear combinations of them as a basis as long as the combinations are all independent vectors.