$H^1(M,\mathbb{Z}_2)$: 1st Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure

$$H^1(M,\mathbb{Z}_2)$$ specifies the 1st cohomology class of manifold $$M$$ (can be regarded as spacetime) with $$\mathbb{Z}_2$$ coefficient,

it is often to see that we say the 1st Stiefel Whitney class

$$w_1 \in H^1(M,\mathbb{Z}_2), \tag{1}$$

However, it looks to me that when we discuss the spinor bundle (associated to fermion in physics), we also say that the fermion line is related to $$H^1(M,\mathbb{Z}_2).$$ For example, on the $$d$$-torus, we need to specify the spin structure on the torus (a spin manifold, with possibly the choice of $$H^1(M,\mathbb{Z}_2)=H^1(T^d,\mathbb{Z}_2)=\bigoplus_d\mathbb{Z}_2$$ along each $$S^1$$-circle direction.) In other words, we may say that roughly

$$\text{periodic and anti-periodic boundary condition of fermions} \in H^1(M,\mathbb{Z}_2),\tag{2}$$

Moreover, one way to understand the fermion moving along a 1-dimensional sub dimensional manifold, may be the spin bordism group generator, say $$\eta$$, where

$$\Omega^{1,Spin}(pt)=\mathbb{Z}_2,\tag{2},$$

where $$\eta$$ on a nontrivial 1-dimensional manifold generates the bordism group.

My question: So how are these eq.(1), (2), (3) are related? Or how are they not related? Can one clarify these with examples?

The set of isomorphism classes of spin structures on a spin manifold $$M$$ is a torsor over $$H^1(M, \mathbb{Z}_2)$$; what this means is that $$H^1(M, \mathbb{Z}_2)$$ acts freely and transitively on it, so the two can be identified but not canonically. An identification requires a choice of spin structure to act as the "origin." The $$n$$-torus happens to have a close-to-canonical choice of spin structure given by the one coming from its Lie group framing, so one can make a close-to-canonical identification in that case.