$\operatorname{P}$ vs. $P$ in probability Below are three separate notations my textbook uses:

$\operatorname{P}(X)$: the probability that an event $X$, which is a subset of the sample space, occurs
$\operatorname{P}(X=x)$: the probability the random variable $X$ assumes the specific numerical value $x$
$P(x)$: probability distribution function, where $P(x)=\operatorname{P}(X=x)$

I understand that it’s generally claimed that upright font is meant to indicate operators while slanted or italicizes font is meant to indicate variables and functions, but in reality, the conventions are much more nuanced than that.
So, why the care to distinguish $\operatorname{P}$ from $P$? How important is it to do so?
Is this simply a regional convention? I have noticed that Europeans use upright symbols more than do Americans (e.g., $\mathrm{d}x$ vs. $dx$), which could account for what I’m seeing.
If you’re curious, my textbook is Mathematics for the International Student: Mathematics HL (Core) third edition by Haese et al. and is Australian in origin.
 A: Your first two examples of notation are really the same, since $\{X=x\}$ is an event. It makes sense that your text takes care to distinguish the probability function $\operatorname{P}(\cdot)$ from the distribution function $P(\cdot)$, since the former operates on sets (events) while the latter operates on real numbers (elements of the sample space). In most treatments of probability that I've encountered, the ambiguity is avoided by having the symbol $P$ (in whatever font or face) reserved just for the probability function, while some other notation ($f$ or $f_X$ or $p$, etc) is used for the distribution function.
A: $\mathsf P(\,\cdot\,)$ is generally used to indicate the probability function over sets of outcomes (aka events) in a probability space, usually denoted as $(\Omega, \mathcal F, \mathsf P)$ or such.  (Where $\Omega$ denotes the sample space, and $\mathcal F$ denotes the sigma-algebra.)  As such, $\mathsf P(X=x)$ is the probability for the event where random variable $X=x$, and $\mathsf P(X\leq x)$ is the probability for the event where $X$ is at most $x$, and so forth.
$$\mathsf P:\Omega\mapsto [0;1]$$
$P(\,\cdot\,)$, or more often $P_X(\,\cdot\,)$ or $p_{\small\lower{0.25ex} X}(\,\cdot\,)$, is typically used to indicate the probability mass function over values of a discrete random variable.   So thus $P(x)$ is the probability mass for the random variable having value $x$.
$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad{ P:\Bbb R\mapsto[0;1]\\(\text{for real-valued discrete random variable, }X)}$$
  In short: $P(x)=\mathsf P(X{=}x)$.
How important is it to make the visual distinction between the two functions?   Well, they are different things, so to be properly rigorous , a distinction should be made.   And although it does not necessarily have to be made by the font, it is convenient to follow the convention.
$\tiny\text{But it is not like the secret maths police will sent out assassins if you forget.}$
