Let's look at the case of $X \sim Pois(\lambda)$.
Since $k \in X(\Omega)$, it is clear that $k^2 \in X(\Omega)^{2}$. Following this logic and from an intuitive view I'd say $\mathbb E[X^{2}]=\sum_{k^{2} \in X(\Omega)^{2}}k^{2}P(X^{2}=k^{2})$, however, I have seen that instead:
$\mathbb E[X^{2}]=\sum_{k \in X({\Omega})}k^{2}P(X=k)$. Can someone explain this notation to me, so that I can understand why the first mentioned is wrong and the latter correct.
If $X(\Omega)$ is uncountable then both notations are wrong because uncountable sums are not defined.
If $X(\Omega)$ is countable then both notations cannot be classified as "wrong" but as tending to redundancy, and both notations should not be used.
Note that here $P(X^2=k^2)=P(X=k)$ so that the summations will have the same outcome.
It is tricky to write $X(\Omega)^2$ because this can also be a notation for the set $X(\Omega)\times X(\Omega)$.
(You most probably want it to be defined as $\{X(\omega)^2\mid\omega\in\Omega\}$.)
Further it is somewhat redundant to use $X(\Omega)^2$ as a set over which $k^2$ ranges, because it is clear allready that for every $k^2\notin X(\Omega)^2$ evidently $P(X^2=k^2)=0$ so that the term on forehand is excluded.
The same objection counts for the use of $X(\Omega)$.
If a random variable is discrete then we always can find a countable set $S\subseteq \mathbb R$ with $P(X\in S)=1$ and for notation of expectations we can practicize the more handsome notation: $$\mathbb Ef(X)=\sum_{k\in S}f(k)P(X=k)$$
Also it is possible to find a minimal $S$ that does that job, and we can identify this $S$ as the support of the discrete random variable.