In a prior post, I developed an "unfolded" gamma distribution generalization of a normal distribution as an example of how to relate a gamma distribution to a normal distribution. This yielded
$$ \text{ND}(x;\mu,\sigma^2,a) = \dfrac{a e^{-2^{-\frac{a}{2}} \left(\frac{1}{\sigma }\right)^a \left| x-\mu\right| ^a}}{2 \sqrt{2} \sigma \Gamma \left(\frac{1}{a}\right)} \,,$$
where the mean is $\mu$, the variance is $\sigma^2$, and the shape is $a>0$, where $a=2$ for an ordinary normal distribution.
This appears to be a different distribution from the generalized error distribution
$$\text{GED}(x;\mu,\alpha,\beta)=\frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(|x-\mu|/\alpha)^\beta}\,,$$
where $\mu$ is the location, $\alpha>0$ is the scale, and $\beta>0$ is the shape and where $\beta=2$ yields a normal distribution. It includes the Laplace distribution when $\beta=1$. As $\beta\rightarrow\infty$, the density converges pointwise to a uniform density on $(\mu-\alpha,\mu+\alpha)$.
Questions Are these distributions the same? If so, how does one convert between them? If not, and $\text{ND}(x;\mu,\sigma^2,a)$ is a different distribution, what are its other properties, for example, does it reduce to a distribution other than a normal distribution?