Suppose we have a probability space $(\Omega, \mathcal{F}, P)$.
A random element is any (measurable) function
$X:\Omega\rightarrow S$ where $S$ is some measurable space.
Depending on which space $S$ we are working with, we get random variables, random vectors, random processes...
For example, if $S=\mathbb{R}^n$, then $X:\Omega\rightarrow\mathbb{R}^n$ is an n-dimensional random vector.
If $S=\mathbb{R}$, then $X:\Omega\rightarrow\mathbb{R}$ is (real) random variable. If $S=\mathbb{C}$ we would have a complex random variable.
In this terminology, we assume that a random variable is a random element which takes values from some scalar field (eg. $\mathbb{R}$). In that sense, a random variable is always scalar, thus same as a scalar random variable.
However, some sources define a random variable to refer to what I defined here as a random element, so in that case a scalar random variable would be what I call here just a random variable - a function on the probability space that takes only scalar values.
$\Omega =\{0,1\}, \mathcal F=\{\emptyset, \Omega\}, X(0)=0,X(1)=1$ is an example since $(X=0) \notin \mathcal F$.
Best Answer
This may be an instance where the most possible general definition gives more insight. Consider a pair $(\Omega, \Sigma)$ where $\Omega$ is the sample space and $\Sigma \subseteq 2^\Omega$ is a $\sigma$-algebra. The measurable sets (i.e., elements of $\Sigma$) are called events. A random variable is a measurable function $X\colon \Omega \to \Bbb R$. By measurable function we mean that for every open interval $]a,b[\subseteq \Bbb R$, we have that $X^{-1}\big(]a,b[\big) \in \Sigma$ is measurable. To put it simply:
A random variable is a nice function from the sample space to the real line (and a random vector takes values in some $\Bbb R^k$, etc.). An event is a certain subset of the sample space, to which a probability may be assigned.
This does not depend on any choice of probability. More precisely, a probability space is a triple $(\Omega, \Sigma, \Bbb P)$ where $(\Omega, \Sigma)$ is as above and $\Bbb P\colon \Sigma \to \Bbb R_{\geq 0}$ is a measure with $\Bbb P(\Omega) = 1$. Then finding $\Bbb P(X = x)$ means finding the "measure" of the event $\{\omega \in \Omega \mid X(\omega) = x\}\in \Sigma$, inside of the sample space $\Omega$.
Example: consider the experiment of tossing a fair coin $n$ times. The sample space is $\Omega = \{{\rm heads}, {\rm tails}\}^n$. The $\sigma$-algebra of events will be all possible subsets of $\Omega$. Say that we write $0$ for heads and $1$ for tails, and the random variable $X \colon \Omega \to \Bbb R$ gives the result of the second toss. That is: $X\colon \Omega \to \Bbb R$ is given by $X(\omega_1,\ldots, \omega_n)= \omega_2$. The event "getting heads on the second toss" is $X^{-1}(0) = \{ (\omega_1,\ldots, \omega_n) \in \Omega \mid \omega_2 = 0 \}$, and the probability of this happening is $\Bbb P(X = 0) = 1/2$.