In a paper I am reading, the following framework was given: Let $S$ be a spinor bundle, over a Riemannian manifold $M$, with Clifford action $$ c:S \otimes \Omega^1(M) \to S. $$ Moreover, let $E$ be a line bundle over $M$ with connection $\nabla$.

The author then speaks of the *canonical* Dirac operator $D$ on $S \otimes E$. What does he mean by this? My guess is as follows: Let $s \in S$ and $e \in E$, such that $\nabla(e) = \sum_i e_i \otimes \omega_i$, for $\omega_i \in \Omega^1(M)$. Moreover, let $D_S$ be the Dirac operator on $S$. I would define $D$ by
$$
D(s \otimes e) = D_S(s) \otimes e + \sum_i c(s \otimes \omega_i) \otimes e_i.
$$
Is this correct? If so, how does one define the Clifford action for $S \otimes E$. Finally, does this work for a twisting by any vector bundle?