The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty_c(\Omega)$ functions.

The third is in general different:

If you let $d = 1$ and $\Omega = \mathbb{R}\setminus \{0\}$, you see that $H_{\Omega}(\Omega) = H^1(\mathbb{R}) \supsetneq H^1_0(\Omega)$. You can create similar examples in higher dimensions (by omitting a hyperplane instead of a point). On the other hand, if $\Omega$ is a Sobolev extension domain (see e.g. Leoni's First Course in Sobolev Spaces) then $u\in H^1_0(\Omega) \iff$ extending $u$ by zero to the exterior gives an $H^1(\mathbb{R}^d)$ function. And in that case the third is also equivalent to the first two.

The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty_c(\Omega)$ functions.

The third is in general different:

If you let $d = 1$ and $\Omega = \mathbb{R}\setminus \{0\}$, you see that $H_{\Omega}(\Omega) = H^1(\mathbb{R}) \supsetneq H^1_0(\Omega)$. You can create similar examples in higher dimensions (by omitting a hyperplane instead of a point). On the other hand, if $\Omega$ is a Sobolev extension domain (see e.g. Leoni's First Course in Sobolev Spaces) then $u\in H^1_0(\Omega) \iff$ extending $u$ by zero to the exterior gives an $H^1(\mathbb{R}^d)$ function. And in that case, the third is equivalent to the first two.