First there can be left tails and right tails, then long tails and short tails. A short-tailed distribution can be thought of a having a finite range, called its support. And a long tail has semi-infinite support in that direction. For right-tail heaviness one does a comparison of either survival functions (RVs) or complimentary cumulative density functions (1-CDF), typically by examining the logarithm of ratio of two different distributions. In general, heavy tailed means heavier than the exponential distribution, and light tailed means lighter than that. A subset of heavier-tailed distributions are called "fat-tailed." From a historical perspective it is most likely that the concept of a fat tail relates to the type I Pareto distribution, i.e.,
\begin{equation}\label{eq:PD}
\text{PD}(t; \alpha, \beta)=
\dfrac{\alpha}{t} \left(\dfrac{\beta}{t}\right) ^{\alpha } \theta(t-\beta)\;,
\end{equation}
where $\alpha$ is the shape parameter, $\beta$ is a scale parameter and $\theta(\cdot)$ is the unit step function such that $\theta(t-\beta)$ is the unit step function time-delayed by $\beta$, and is used to make a product that is non-zero only when $t> \beta$.
From Juran, "The Pareto Principle (Sic, 80-20 rule) gets its name from the Italian-born economist Vilfredo Pareto (1848-1923), who observed that a relative few people held the majority of the wealth (20%) – back in 1895. Pareto developed logarithmic mathematical models to describe this non-uniform distribution of wealth and the mathematician M.O. Lorenz developed graphs to illustrate it."
Next, let us consider the attitude toward wealth at that time. Renzaho quotes Grivetti saying "At the turn of the 20th century in North America obesity was admired; wealthy consumers exhibited their wealth around their waist. Fat cheeks and ample stomachs were visual cues that individuals were healthy, not infected with the dreaded slim tuberculosis. Photographs of American executives taken during the late 19th and early 20th centuries reveal that dietary intakes of wealthy gentlemen regularly exceeded calories expended."
The history of those times lends considerable weight to those words. From the US 1910 census, tuberculosis, A.K.A, "consumption," which literally eats the body from the inside out, had been endemic for decades and was responsible for approximately 15 deaths per 1,000 inhabitants per year, or, if you wish, approximately 30 times the annual death rate from SARS-CoV-2. Thus, the more modern concept of slimness being healthy was not plausibly in vogue.
Next, the term "fat cats" came to describe wealthy political donors in approximately 1920 or earlier and Pareto's work was first translated into English in 1916. Wesolowski et al. summarize the prevailing attitudes of those times in a footnote, "Ironically, the fat-tailed distribution of wealth inspired Karl Marx’s [49], as well as Benito Mussolini’s economic policies as diametrically opposite and extreme reactions to the same statistics [50]." and attribute the meaning of fat tails as follows, "The statistical form for a power law is the Pareto distribution (PD), which like the Cauchy distribution, has tails so heavy they confer unusual statistical properties and have been given the name fat-tailed distributions. Power laws are scale independent and intrinsically fractal."
In that paper, [49] refers to Of Fat Cats and Fat Tails: From the Financial Crisis to the ‘New’ Probabilistic Marxism and [50] refers to Pareto and Fascism Reconsidered by Zanden. That Vilfredo Pareto had Benito Mussolini as his most well known student should be lost on no one.