Let $dS_t = \mu_tS_tdt + \sigma_tS_tdW_t$ be the underlying GBM (Geometric Brownian Motion)-like dynamics as in the question.
Let $B_t$ a Brownian motion such that $d[B,W]_t = \rho dt$, $\rho\in[-1,1].$
CIR (Cox-Ingersoll-Ross) for $\sigma_t^2$ (when combined with GBM-like underlying dynamics, it is the popular Heston SV model)
$$d\sigma_t^2 = \kappa(\theta - \sigma_t^2)dt + \zeta \sigma_tdB_t$$
GBM for $\sigma_t^2$ and $\rho=0$ (when combined with GBM-like underlying dynamics, it is the Hull-White SV model)
$$d\sigma_t^2 = -\kappa\sigma_t^2dt + \zeta \sigma_t^2dB_t$$
(Yours) Exponential OU for $\sigma_t$ (when combined with GBM-like underlying dynamics, it has no special name)
$$d\ln \sigma_t = \kappa(\theta - \ln \sigma_t)dt + \zeta dB_t$$
Lognormal for $\sigma_t$ (when combined with GBM-like underlying dynamics, it has no special name)
$$d\sigma_t = \kappa(\theta - \sigma_t)dt + \zeta \sigma_tdB_t$$
They can all be useful, depending on what you want or, simply, as benchmark of one against the other (when calibrated to the same targets). Note that if you are willing to revisit the underlying dynamics itself, you get other SV (stochastic volatility) models. For example, look up SABR or SLV (stochastic local volatility) models.