How these two approaches to spinors in curved spacetimes relate? Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a vielbein $e^a_\mu$. In that setting a Dirac spinor is a $\Psi(x)\in\mathbb{C}^4$ with some properties.

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*It transforms under the Local Lorentz Symmetry as $$\Psi'(x)=L(\omega(x))\Psi(x),\quad L(\omega(x))=e^{\frac{i}{2}\omega^{ab}(x)\Sigma_{ab}},\quad \Sigma_{ab}=\dfrac{1}{4}[\gamma_a,\gamma_b],$$
where $\gamma_a$ are the standard flat spacetime gamma matrices.


*It can be covariantly differentiated as $$D_\mu\Psi(x)=\partial_\mu\Psi(x)+\dfrac{i}{2}B_\mu^{ab}\Sigma_{ab}\Psi(x)$$
On the other hand there is another approach which is more rigorous and relies on spin structures. In that sense we start with the bundle of orthonormal frames $\pi_F:{\cal F}(M)\to M$ and define a spin structure to be a principal ${\rm SL}(2,\mathbb{C})$-bundle $\pi_P:P\to M$ together with a principal map $\Phi:P\to{\cal F}(M)$ such that if $\rho:{\rm SL}(2,\mathbb{C})\to {\rm SO}(1,3)$ is the covering map then $\Phi(e\cdot g)=\Phi(e)\cdot \rho(g)$. Now a Dirac spinor will be a section of the associated bundle to $P$ constructed from the Dirac representation of ${\rm SL}(2,\mathbb{C})$.
In this second approach one has to discuss whether a spin structure exists, the answer being that it exists if and only if the second Stiefel-Whitney class of $M$ is zero. A second question is whether the spin structure is unique and I'm aware that in some cases it is not. Now, to be honest I have not seem this spin structure definition being used in practice, so I know little more than the definition.
My question here is: how does these two approaches relate? The approach in the lecture notes seems way easier to use in practice, but I can't see where the spin structure lies in there. In particular it is not clear to me where the map $\Phi$ lies in there and how distinct spin structures may appear. Still I have the impression that we can start from the spin structure and reach a point in which to work in practice we get to the approach in the lecture notes.
 A: The two definitions are equivalent once we think about what is required for the spinor field from the first to be mathematically well-defined:
If you want to say what sort of object the spinor field $\Psi$ is mathematically, it has to be a section of a (complex) vector bundle $S\to M$ with fiber $\mathbb{C}^4$ such that there is a consistent action of the spin group at every point. Since $L(\omega(x)) \in \mathrm{SO}(S_x)$ - the spin transformation acts as a special orthonormal transformation at every point of the bundle $S$ - this means we should consider a bundle with the spin group as a fiber that projects down onto the special orthonormal frame bundle of $S$: $p : P_\text{Spin}(S) \to P_\text{SO}(S)$ with $p(lg) = p(l)\pi(g)$ where $l\in P_\text{Spin}, g \in \text{Spin}(S)$ and $\pi : \text{Spin}(S) \to \mathrm{SO}(S)$ is the double covering.
Furthermore, physically, we want that the transformation of $\Psi$ happens "concurrently" to that of a vector - a rotation rotates both vectors and spinors, there are not different kinds of rotations that would act separately. So the bundle $P_\text{Spin}(S)$ above must also project down onto the special orthonormal frame bundle $P_\text{SO}(TM)$ of the manifold itself. $P_\text{Spin}(S)$ together with that projection is now the spin structure from your second definition.
Conversely, the spin structure from the second definition yields the spinor bundle $S$ and hence spinor fields as its sections via the usual associated bundle construction.
