Is $H^{1}(\Omega)$ defined by $\left\{u \in L^{2}(\Omega) | \nabla u \in L^{2}(\Omega)^{d}, u=0 \text { on } \partial \Omega\right\}$

Question

I can't see the difference between these two Sobolev spaces:

$H^{1}(\Omega):=\left\{u \in L^{2}(\Omega) | \nabla u \in L^{2}(\Omega)^{d}, u=0 \text { on } \partial \Omega\right\}$

and $H_{0}^{1}(\Omega)=\left\{v \in H^{1}(\Omega) | v_{| \partial \Omega}=0 \text { on } \partial \Omega\right\}$

Where $\gamma_{0}$ is linear from $H^{1}(\Omega) \cap C(\bar{\Omega})$ to $L^{2}(\partial \Omega) \cap C(\overline{\partial \Omega})$ by $\gamma_{0} v:=v_{| \partial \Omega}$

the definition that I usually use is:

$H^{1}(\Omega)=\left\{u \in L^{2}(\Omega) | D_{i} u \in L^{2}(\Omega), \forall i=1, \ldots, N\right\}$

but in the paper I am reading the authors state that:

$H^{1}(\Omega):=\left\{u \in L^{2}(\Omega) | \nabla u \in L^{2}(\Omega)^{d}, u=0 \text { on } \partial \Omega\right\}$

I am a beginner in Sobolev spaces.. someone can give me a hint?