Given $n$ qubits that reside in an Euclidean space $\mathcal E$ of dimension $2^n$, a quantum code is a subspace of $\mathcal E$.
Note then that a quantum code is just a subspace of a larger space, and we have not mentioned anything about the definitions of its properties pertaining to error-correction.
Here, a stabilizer $\mathcal S$ refers really to an abelian subgroup of $\mathbb F_4^n$. In this sense, the words,
element' andgenerator' just inherit the definitions from group theory.
The connection between operator theory and this discrete set $\mathcal S$ is well known, and can be found in Daniel Gottesman's thesis on the arxiv.
If you'd like a formal operator theory introduction to quantum codes, there are many papers you can find, but you can for example find a very terse introduction in my paper: https://arxiv.org/abs/1604.07925
Do note that not all quantum codes are `stabilizer codes'.
The toric code can be understood to be a special stabilizer code, where the stabilizer is generated by elements of $\mathbb F_4^n$ of weight 4.