In many places one can find the statement that the existence of of a Spin-structure on a (compact) Kähler manifold $M$ of (complex) dimension $n$ is equivalent to the existence of a $\theta$-characteristic, i.e. a line bundle $L$ such that the canonical line bundle satisfies $$\Omega^{n,0}\cong L\otimes L.$$
I am used to the existence property in terms of the vanishing of the second Stiefel-Whitney class $w_2(M)$. Does the proof simply follow from the equalities (which follow from the properties of the first Chern class) $$c_1(L)+c_1(L)=c_1(L\otimes L)=c_1(\Omega^{n,0})=-c_1(T_\mathbb{C}M)$$ and $$w_2(M)=c_1(M)\bmod2?$$
Whilst trying to understand this I also got confused by the nLab-page "https://ncatlab.org/nlab/show/canonical+bundle", where they claim that the $\theta$-characteristic satisfies $$c_1(\Omega^{n,0})=2\theta=\theta\cup\theta.$$ But since the first Chern class of line bundles is additive with respect to tensor products and the cup product would give a class in $H^4(M;\mathbb{Z})$ and not $H^2(M;\mathbb{Z})$, this last equality cannot be correct, right?