Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$, let $L^2$ denote all (equivalence classes for a.s. equality of) random variables $X$ such that $\mathbb{E}\{X^2\}<\infty$. We henceforth identify all random variables $X$, $Y$ in $L^2$ that are equal a.s. and consider them to be representatives of the same random variable. This has the consequence that if $E\{X^2\}=0$, we can conclude that $X=0$ (and not only that $X=0$ a.s.).
I am not sure I am correctly interpreting the above-quoted statements.
What I understand is that, since $E\{X^2\}=0$, given that $\text{Var}(X)=E\{X^2\}-E\{X\}^2\geq0$, it must hold that $E\{X\}=0$. So, at this point one can say that, since $E\{X\}=0$ and $\text{Var}(X)=0$, $X$ follows a degenerate distribution, that is that $X=0$ a.s..
Now, pretending that what I have said so far is ok, does the statement "we can conclude that $X=0$ (and not only that $X=0$ a.s.)" follow from the fact that (see part in bold) "$L^2$ denotes all (equivalence classes for a.s. equality of) random variables $X$ such that $\mathbb{E}\{X^2\}<\infty$"?
If my reasoning was totally wrong, could you please clarify the meaning of the paragraph above?
This is actually way more fundamental and only a matter of perspective. $X = 0$ in $L^2$ has a different meaning than $X = 0$ in the space of random variables.
Before introducing $L^2$ we knew that $E\lbrace X^2 \rbrace = 0 \iff X = 0$ a.s..
Now things have changed as $L^2$ only contains equivalence classes (i.e. sets) and is a completely different Vector space than the space of random variables.
Now, what does it mean for $[X] \in L^2$ to be $0 \in L^2$ ($[X]$ denotes the equivalence class of some random variable $X$)? It means that $Y = 0$ a.s. for all $Y \in [X]$ by definition of $L^2$. So: The zero element in $L^2$ is just not a random variable anymore, it's a completely different object.
Remark: We can still write down integrals of equivalence classes as the integral of an arbitrary representant, because every random variable in an equivalence class has the same integral (because they only differ on sets with measure zero). Often the goal of this new perspective is to introduce a norm $\lVert X \rVert_{L^2} := \sqrt{E\lbrace X^2 \rbrace}$ on $L^2$ which requires $\lVert x \rVert_{L^2} = 0 \iff x = 0$ in $L^2$.
This is the concept of so called quotient spaces about which you can learn in a good Linear Algebra book.