The lemmas leading up to the definition are correct, but the precise definition switches the functors $\textrm{--}^G$ and $\textrm{--}_G$. The correct definition should be:
$$H_{n}(G,A) = L_{n}(\textrm{--}_{G})(A) \cong Tor_{n}^{{\mathbb{Z}G}}(\mathbb{Z},A)$$
and similarly for cohomology.
The lemmas leading up to the definition are correct, but the precise definition switches the functors$\textrm{--}^G$ and $\textrm{--}_G$ . The correct definition should be:
and similarly for cohomology.