What is the difference between a probability distribution on events and random variables?

For the purpose of simplicity, assume everything below is only in the discrete domain.

A $\text{probability space}$ is usually defined as a triple $(\Omega , 2^\Omega , P)$ where

$\Omega := \text{set of outcomes}$

$2^\Omega:= \text{set of events (for simplicity)}$

$P:= \text{mapping from } 2^\Omega \mapsto [0,1] \text{ which assigns a probability to each event}$

My first question is whether $P$ is a probability distribution or a probability measure, and what exactly is the difference between these two ideas.

Moreover, I am trying to understand where the concept of a $\text{random variable}$ fits in with all this.

From my reading a random variable (over the probability space above) $X:= \text{mapping from } \Omega \mapsto \mathbb{R}$.

When we say $P(X=a)$, is this $P$ referring to the same probability distribution of the space? What exactly is the object $X=a$? I would think it refers to the set of all outcomes in $\Omega$ that map to $a$ but my reading does not elucidate this.

What is the ultimate confusion is the statement: $\text{we will use the notation} P(X) \text{ to denote the distribution of the random variable X}$

Again, what exactly is meant by the probability distribution $P$ here? Does it also refer to the $P$ of the probability space that $X$ is defined over? How is this distribution different?

Probability distribution and probability measure are synonyms.

$[X=a]=X^{-1}(\{a\})=\{\omega\in\Omega\mid X(\omega)=a\}$ hence $P(X=a)=P(X^{-1}(\{a\}))$.

The distribution of the random variable $X:\Omega\to\mathbb R$ is the unique probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by $\mu(B)=P(X\in B)$ for every $B$ in $\mathcal B(\mathbb R)$, where $[X\in B]=X^{-1}(B)=\{\omega\in\Omega\mid X(\omega)\in B\}$.