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:

1. Ledoux and Talagrand, *Probability in Banach Spaces*, p. 10 and chapter 11

2. Klartag, "[Uniform almost sub-gaussian estimates for linear
functionals on convex sets][1]"

3. Giannopoulos, Pauoris, and Valettas, "[$\Psi_\alpha$-estimates for marginals of log-concave probability measures][2]"


  [1]: http://www.math.tau.ac.il/~klartagb/papers/psitwo.pdf
  [2]: http://www.math.tamu.edu/~grigoris/log-concave-marginals.pdf