In his book Algebraic Geometry, Hartshorne introduces the twisting sheaf on a scheme $X = \operatorname{Proj} $ of a graded algebra $S$. He does this by taking the associated sheaf on $X$ of the $S$-module $S(n)$. This is on pg. 117 of my edition.
I don't know what the module $S(n)$ is. Is it the submodule of $S$ generated by the component of homogeneous elements of degree $n$?
As stated in the comments, the definition is $S(n)_d = S_{n+d}$. For your reference, the definition is on page 50 in Chapter I section 7. (This is not the same as the module generated by $S_n$ mentioned in the question, even in the ungraded sense. In special cases such as regular module over a polynomial ring, it is.)