One often denotes $\psi_\alpha (x) = e^{x^\alpha} - 1$ for $\alpha \ge 1$, and the corresponding Orlicz space is $L_{\psi_\alpha}$. An estimate on the $L_{\psi_\alpha}$ norm is also often referred to simply as a $\psi_\alpha$ estimate. Of course, the case $\alpha = 2$ which you asked about is the one most often of interest.
One might also call this space $\exp(L^\alpha)$, in analogy with $L^p$ and $L\log L$, etc., but I've never seen this done.
Added: Here are a few references where the $\psi_\alpha$ and $L_{\psi_\alpha}$ notations are used:
Ledoux and Talagrand, Probability in Banach Spaces, p. 10 and chapter 11
Klartag, "Uniform almost sub-gaussian estimates for linear functionals on convex sets"
Giannopoulos, Pauoris, and Valettas, "$\Psi_\alpha$-estimates for marginals of log-concave probability measures"