I am writing to add some necessary details to the proof of the statement $\textbf{2}$.
The OP didn't provide his definition of $\textit{pole}$. By inspecting his proof, I guess his definition of $\textit{pole}$ is based on the singular part of Laurent series. To be more specific, his definition might be that, if the Laurent series at a singularity $z_0$ has at least one and at most finitely many terms of negative powers, then $z_0$ is called a $\textit{pole}$.
So in the proof of the converse direction of the statement $\textbf{2}$, we have to show that the Laurent series of $f$ at $z_0$ has at least one and at most finitely many terms of negative powers.
Proof:
First let's suppose that $\text{lim}_{z\to z_0}f(z)=\infty$.
Let $g(z):=\frac{1}{f(z)}$. Then $g(z)$ is well-defined in a ball centered at $z_0$ and $\text{lim}_{z\to z_0}g(z)=0.$ This implies that $g$ is holomorphic in the ball.
Since $g$ is holomorphic in the ball and has a zero at $z_0$, we can find some holomorphic function $h$ s.t. $g(z)=(z-z_0)^k h(z)$ with $h(z_0)\neq0$, where $k\ge 1.$
Since $h$ is holomorphic in the ball and $h(z_0)\neq0$, we may assume $h$ is nonzero in the ball. (Because we can always shrink the ball to a smaller ball, it doesn't really matter.) So $\frac{1}{h}$ is holomorphic in the ball and hence we can write $\frac{1}{h(z)}=\sum_{n=0}^\infty a_n(z-z_0)^n$ for some coefficients $\{a_n\}$ by the Taylor theorem.
So at every point $z$ in the punctual ball (i.e. the ball excluding $z_0$), we have $$f(z)=\frac{1}{g(z)}=\frac{1}{(z-z_0)^k h(z)}=\sum_{n=0}^\infty a_n(z-z_0)^{n-k}=\sum_{n=-k}^\infty a_{n+k}(z-z_0)^n.$$
By the definition, we conclude that $z_0$ is indeed a pole.
$$\tag*{$\blacksquare$}$$
Best Answer
In general, one should remember that anybody can edit a Wikipedia page, regardless of their qualification. In this case, by looking through the history of edits, this particular offending sentence (or, rather, some version of it) was initially written by an engineer, who clearly did not understand the definition; then another engineer tried to correct it but made it even worse. The correct sentence should be something like:
"In the language of topology, an isolated singularity of $f$ is an isolated point of the set $\partial \Omega$."
or
"In the language of topology, an isolated singularity of $f$ is an isolated point of the set ${\mathbb C}\setminus \Omega$."
(There is some disagreement if $\infty$ is allowed as an isolated singularity; if it is, then $\partial \Omega$ is understood as the boundary of $\Omega$ in the Riemann sphere.)