How to interpret these asymptotic runtime bounds for discrete logarithm algorithms?
The confusion is that in the expression for $L(\alpha)$, $n$ is size of the field, whereas in $n^{O(\log n)}$, $n$ is the size of the input. The correct comparison is between $L(\alpha) = \exp O((\log n)^\alpha (\log\log n)^{1-\alpha})$ and $(\log n)^{O(\log\log n)} = \exp O((\log\log n)^2)$, which is indeed much smaller.

Note that your suggestion to plot $L(\alpha)$ for $\alpha = 1/4 + O(1)$ and $\alpha = 1/4 + O(n)$ doesn't make much sense; you should think of $\alpha$ as a constant. It could make sense to consider $\alpha = 1/4 + o(1)$ (or $\alpha = 1/4 + \epsilon$, where $\epsilon$ is understood to be "small"), though it's not clear how exactly you'd plot it. Indeed, there is no reason to plot, and a plot wouldn't make too much sense without knowing the hidden big O constant. In practice you just do experiments.