In probability theory, the term *subexponential distribution* has historically been used for a distribution whose CDF $F(x)$ satisfies the relation
$$
n(1-F(x)) \sim 1 - F^{*n}(x)
$$ for any $n \ge 1$ as $x \to \pm \infty$ (depending on the context), where $F^{*n}(x)$ is the CDF of the $n$-fold additive convolution. The interpretation is that for large $x$, $F(x) \approx 1-\epsilon$ and so $$
n(1-(1-\epsilon)) = n\epsilon \approx 1 - (1-\epsilon)^n
$$ and the latter represents the maximum of $n$ independent copies of the distribution, so we can say that $$
\mathbb{P}[X_1 + X_2 + \ldots + X_n > x] \sim \mathbb{P}[\max\{X_1,X_2,\ldots,X_n\}>x]
$$ for large $x$, or specifically that it is a distribution such that if the sum of $n$ independent copies is large, it is likely due to the contribution of one particularly large jump. So *subexponential* is a type of large-tailed distribution, subject to extreme events.

On the other hand, a subgaussian distribution is one whose moment generating function does not grow faster than the Gaussian-like function, so
$$
\mathbb{E}[e^{\lambda(X-\mathbb{E}[X])}] \le e^{\lambda^2\sigma^2/2}
$$ for all $\lambda$ and some $\sigma^2 >0$. This guarantees tails that are no larger than Gaussian. If we relax this so that it only holds in a neighborhood of zero, specifically that
$$
\mathbb{E}[e^{\lambda(X-\mathbb{E}[X])}] \le e^{\lambda^2\sigma^2/2}, \ \ \ \ \ |\lambda| < 1/\alpha
$$ then the tail bounds become
$$
\mathbb{P}[|X-\mathbb{E}[X]|>t] \le 2e^{-t/2\alpha}, \ \ \ \ \ t> \sigma^2/\alpha
$$ It seems that in some circles, these distributions are now being referred to as *subexponential*. See, for instance, https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/4notes.pdf, and other top hits on "subexponential distribution".

Frustratingly, these conditions are quite opposite of one another: one implies heavy tails, the other light tails. The latter seems more appropriate, meaning "tails no heavier than exponential". My question is, what is the historical background that led to calling the former subexponential?