You are using the word "domain" to mean two mathematically distinct concepts, and this is confusing you. Per your other (very similar) question asked around the same time, I believe what you are really interested in is the support of the distribution of the random variable.
To try to explain things more concretely, let us imagine using a random number generator to simulate a normal distribution. Assume the random number generator produces a uniformly random real number $X$ in $[0,1]$. Then you can apply a suitable function $f\colon [0,1]\to\mathbb R$ to $X$ to obtain a standard normal random variable. (If you want instructions of how to explicitly write down such a function, see https://en.wikipedia.org/wiki/Inverse_transform_sampling)
Now the incredible (or incredibly boring) insight of (formal) probability theory is that $f$ is the same thing as the standard normal random variable. In your notation, $\Omega_2=[0,1]$ is the sample space and $f$ is the random variable. But by the same token, I could have started on another domain, like saying $Y$ is uniformly random in $[0,2]$, and found a different function $g$, with the property that $g(Y)$ and $f(X)$ have the same distribution - standard normal in both cases. So we would have $\Omega_2'=[0,2]$. So $\Omega_2$ is not intrinsic to the standard normal distribution - it can be (almost) anything.
Now let's address $\Omega_1$. We all know the standard normal density is a constant times $e^{-x^2/2}$. Of course, it is a function from $\mathbb R$ to $\mathbb R$. But this does not really tell us anything about the standard normal density. The same is true for densities that are not supported on all of $\mathbb R$, like the exponential density, which is $$e^{-x}\cdot 1[x\geq 0].$$
This funny notation means that the density is a function, defined for all $x\in\mathbb R$, that happens to equal $0$ when $x$ is negative. (In particular, it is not continuous at $x=0$.) So in this case too, we can say that the domain of the density is all of $\mathbb R$. But the support is only $[0,\infty)$ - the set of positive real numbers. In the other question you asked, I gave a careful definition of the support. For the purposes of this question, you can think of it as just the set of $x\in\mathbb R$ for which the density is non-zero.