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.