The correspondence between complex spin structures and associated clifford module structure

Question

The usual definition of a $spin^c$ structure on a riemannian manifold $M$ is often given by (the equivalence class of) the lifting of the frame bundle $P\to M$ of the tangent bundle $TM$ via the standard homomorphism $Spin^c(n)\to SO(n)$.

And there is yet another definition given in Monopoles and Three-manifolds, Kronheimer and Mrowka:

A $spin^c$ structure is a choice of "spinor bundle" $S$ (a complex vector bundle of proper dimension) and a Clifford multiplication $\rho : TM\to End(S)$ which satisfies $\rho(X)^2=-\left<X,X\right>id_{S}$ (hence extendable to the action of the Clifford algebra on $S$, $Cl(TM)\to End(S)$ which we furthermore assume that this restricts to the spin representation $Cl(T_xM)\to End(S_x)$ on each fiber).

(It might be the case that this definition is only valid in low-dimensional manifolds, especially 3 and 4Ds.)

Sadly this is where I stuck. After some searching, the equivalence between these two definitions is a forklore to experts, but it is hard to find the reference to the proof. My current understanding is,

1. From the first definition to the second, the correspondence is (should be?) via the associated bundle construction w.r.t. the standard $spin^c$ representation $Spin^c(n)\to End(\Delta)$. For this direction, I have no confusion, this is totally fine with me.
2. For the reverse direction, I have little clue. I only feel there would be a similar construction as "the frame bundle construction of a vector bundle that makes a correspondence between $SO(n)$-bundles and (oriented) vector bundles". But what can be a version of frames in the case of spinors?
3. Or it might be possible that we can directly show the bijectiveness of the correspondence from the first to the second. But, from the second definition, I feel there's more freedom than the first, say we can choose a trivial bundle of proper dimension as $S$ and then there's no obstruction in defining a Clifford multiplication map $TM\to End(S)$ (at least considering the fact that End(S) has much larger dimension than $TM$). So, in this approach, the bijective correspondence seems rather a miracle to me.