Why does Gumbel distribution have two different expressions? Let $X_1,X_2,\dots,X_n$ be iid random variables with distribution function $F(x)$ and $M_n:=\max\{X_1,\dots,X_n\}$. By  the extreme value theorem, there exist two sequences of real numbers $a_n>0$ and $b_n$ such that the following limits converge to a non-degenerate distribution function $G(x)$.
$$
G(x;\mu,\sigma)=P(X\le x; \mu. \sigma)=\exp(-\exp(\frac{x-\mu}{\sigma}))
$$
for $x\in R$.
Q1: Why does Gumbel have two different expressions?

Also, the Gumbel, Frechet, and Weibull are from the family of extreme value distributions:
$$
G_{\gamma}=\exp(-(1+\gamma x)^{-1/\gamma})
$$
for $1+\gamma x>0$.
Why this paper wrote in another version as follows
$$
G(x;\mu, \sigma, k)=\exp(-[1－\frac{k(x-\mu)}{\sigma}]^{1/k})
$$

Quoting from Wikipedia:

In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.

It's not always clear whether the maximum or minimum version of the Gumbel is being used. One is obtained by reversing the sign of $x$ from the other. The version with $\mu$ and $\sigma$ allows for different location and scale parameters than the standard Gumbel.
For the generalized extreme value distribution, the definition of the shape parameter ($\gamma$ or $k$ in your examples) has two different conventions, one the negative of the other. See this page for an example of the resulting confusion.
As to "why" there are different parameterizations, I suppose it has to do with historical patterns of use of minimum versus maximum Gumbel distributions and their extensions in different fields of interest. For example, the standard parameterization of an accelerated-failure-time Weibull survival model is based on the minimum extreme value Gumbel, leading to errors if you try to use the maximum, which is sometimes the generic default.
Those differences are not going to go away. For all of the distributions you note, it's critical to be sure that you know precisely which parameterization is being used.
