In my class notes the Sobolev space is defined in a way I was unaware of. Given an open set $\Omega\subset \mathbb{R}^d$,
$$W^{1,p}(\Omega)=\{ u \in L^1_{\text{loc}(\Omega)} :\text{ admits weak derivative and } u\in L^p(\Omega), \nabla u \in (L^p(\Omega))^d\}.$$
I had never seen this definition, the reference for classes is also the Evans' book, which defines the Sobolev space as
$$W^{k,p}(\Omega) = \{ u \in L^1_{\text{loc}}(\Omega) : D^\alpha u \in L^p(\Omega), |\alpha| \leq k \}.$$
For the case $k=1$ are these two definitions equivalent? Why in this case is it required that $u \in L^1_{\text{loc}}(\Omega)$ and then $u \in L^1(\Omega)$? Maybe it's a typo and it should be $D^\alpha u \in L^p(\Omega)$.
Could anyone clarify this definition for me?
The definition is correct, and the two definitions are equivalent.
However, there is another definition, also equivalent.
First set $$ \hat C^{k,p}(\Omega)=\{u\in C^{\infty}(\Omega : \|u\|_{k,p}<\infty\}, $$ where $$ \|u\|_{k,p}=\left(\sum_{|\alpha|\le k} \int_\Omega |D^\alpha u|^p\,dx\right)^{1/p}. $$ Clearly, $\hat C^{k,p}(\Omega)$ is not complete, with respect to $\|\cdot\|_{k,p}$. Then $W^{k,p}(\Omega)$ is the completion of $\hat C^{k,p}(\Omega)$.
A proof of the equivalence of the definitions can be found for example in Avner Friedman's PDEs.