At least the class still uses a sample space $\Omega$ and still talks about functions $g:\Omega\rightarrow\mathbb{R}$ (putting details about sigma algebras aside, these functions $g$ are what everyone else would call random variables).
I think your class wants to view $W$ as a "factory" (or piece of software) that, when repeatedly run, produces (independent) outcomes. It wants to use $W$ as the name of the software, and then use $w_1, w_2, w_3$ as "realizations" from $W$. It defines $X=g(W)$ as a new piece of software that puts the output of the previous software through $g$. So then "realizations" of $X$ are $g(w_1), g(w_2), g(w_3)$.
This view is a significant departure from standard probability theory where there is a single sample space $\Omega$, and the outcome $\omega \in \Omega$ determines the values $X_1(\omega), X_2(\omega), X_3(\omega), ...$ for all random variables $\{X_1, X_2, X_3, \ldots\}$. One way to make this consistent is if you consider the product space
$$\tilde{\Omega} = \underbrace{\Omega \times \Omega \times ... \times \Omega}_{\mbox{$n$ times}}$$
so a single outcome $\omega \in \tilde{\Omega}$ has the form $\omega = (\omega_1, \omega_2, ..., \omega_n)$ and you can define
$$X_i(\omega_1, ..., \omega_n) = g(\omega_i) \quad \forall \omega\in \tilde{\Omega}$$
The trouble with this is that it implicitly assumes all "trials" or "samples" are always independent. You should ask your professor how to handle cases when the $X_1, X_2, ..., X_n$ random variables can be correlated.
Regarding standard definitions I would use this:
A measurable space is a pair $(\Omega, \mathcal{F})$ where $\Omega$ is a nonempty set and $\mathcal{F}$ is a sigma algebra on $\Omega$.
A probability space is a triplet $(\Omega, \mathcal{F}, P)$ where $(\Omega, \mathcal{F})$ is a measurable space and $P:\mathcal{F}\rightarrow[0,1]$ a function that satisfies the 3 axioms of probability.
Now suppose we have a probability space $(\Omega, \mathcal{F}, P)$:
A random variable is a function $X:\Omega\rightarrow\mathbb{R}$ that satisfies
$$ \{\omega \in \Omega : X(\omega)\leq x\} \in \mathcal{F} \quad \forall x \in \mathbb{R}$$
Equivalently, it can be shown that $X:\Omega\rightarrow\mathbb{R}$ is a random variable if and only if
$$ \{\omega \in \Omega : X(\omega) \in B\} \in \mathcal{F} \quad \forall B \in \mathcal{B}(\mathbb{R})$$
where $\mathcal{B}(\mathbb{R})$ is the standard Borel sigma algebra on $\mathbb{R}$.
Given a measurable space $(V, \mathcal{G})$, a random element is a function $X:\Omega\rightarrow V$ that satisfies
$$ \{\omega \in \Omega : X(\omega) \in B\} \in \mathcal{F}\quad \forall B \in \mathcal{G}$$
Under these definitions, a random variable is just a random element on the output measurable space $(V,\mathcal{G})=(\mathbb{R}, \mathcal{B}(\mathbb{R}))$.
For two measurable spaces $(V_1, \mathcal{G}_1)$ and $(V_2, \mathcal{G}_2)$ and two random elements $X:\Omega\rightarrow V_1$ and $Y:\Omega\rightarrow V_2$, we say $X, Y$ are independent if
$$ P[\{X \in A\}\cap \{Y\in B\}]=P[X\in A]P[Y\in B] \quad \forall A \in \mathcal{G}_1, B \in \mathcal{G}_2$$
Some people call random elements "random variables." I suspect your last question on the definition 2.0.1 uses $\mathcal{X}=V_1$ and $\mathcal{Y}=V_2$, ignores the issue of sigma algebras, and if we assume $X, Y$ are real-valued then indeed you can assume $\mathcal{X}=\mathcal{Y}=\mathbb{R}$. Alternatively, perhaps $\mathcal{X}$ is the subset of $\mathbb{R}$ consisting of values that $X$ can take, that is, $\mathcal{X}=\{X(\omega): \omega \in \Omega\}$ (the image of the random variable). Some people use notation $S_X$ for this.
