I'm wondering about a general way to define the action of time reversal on a fermion field $\psi$. From a few sources I've read (e.g. appendix A of Witten's paper on fermion path integrals), it seems like in general, a reflection of the coordinate $x^\mu$ in space time corresponds to the transformation $$ R_\mu:\psi(x) \mapsto \gamma_\mu \psi(R_\mu(x)),$$ where $R_\mu(x^\nu) = -x^\nu$ if $\mu=\nu$ and $R_\mu(x^\nu)=x^\nu$ otherwise. If this is true, we should be able to choose time reversal so that it acts on fermions as $$T: \psi(t,x) \mapsto \gamma_0 \psi(-t,x).$$ Indeed, I have seen papers (e.g. this paper by Cordova et al) where such a choice is made, at least for theories in 2+1 dimensions.
I'm curious if this is a satisfactory definition of time reversal in general (which seems to be implied by the references mentioned), and if it's not, whether there is a simple modification that makes it into a general definition.
One reason why I don't think it can be general is that it seems to be dependent on the signature we choose. For example, the Dirac mass in 2+1D is supposed to be $T$-odd. If we are in $(-,+,+)$ signature then we can choose e.g. $\gamma_0 = i\sigma^y$, in which case one checks that the Hermitian term $im\bar\psi\psi$ is indeed $T$-odd. However, if we are in $(+,-,-)$ signature where we can choose e.g. $\gamma_0=\sigma^x$, then the Hermitian term $m\bar\psi\psi$ is $T$-even. So clearly something with our definition of $T$ is not right.