I am not sure if the following two types of transcendental names in the literature, so I will assign crudely a temporary moniker. We say a transcendental number $\alpha$ is of type I if for all non-zero algebraic numbers $\beta$, we have that $\alpha^\beta$ is transcendental. We say that $\alpha$ is of type II if there exists a non-zero algebraic number $\beta$ such that $\alpha^\beta$ is algebraic.

For example, it is known that $e$ is of type I, by the Lindemann-Weierstrass theorem. It is also known that there exist transcendental numbers of type II, established for example by the Gelfond-Schneider theorem. Here we can take $\alpha = 2^\sqrt{2}$ for example, and $\beta = \sqrt{2}$.

My question is has there any attempts to study transcendental numbers of type I or type II? To me it seems that transcendental numbers of type I are 'more' transcendental than those of type II. Is there any definition to make this intuition rigorous?