There is not really a standard notion of an operation that takes $\xi\to \xi^*$ for a Grassmann variable and in a Euclidean signature Grassmann path integral $\psi$ and $\psi^*$ are independent objects. I don't know how widespread P&S's $(ab)^*=-b^*a^*$ is but I find it confusing. Also appearences of $i\sigma_2$ as in P&S are dependent on the choice of representation for the Gamma matrices.
I think is much simpler just to take
the Grassmann-variable Majorana action to be
$$
S_{\rm Majorana} ={\textstyle \frac 12} \int \psi^T C (\gamma^\mu \partial_\mu +m)\psi\, d^4x
$$
where $C$ is the charge conjugation matrix defined by $C\gamma^\mu C^{-1}= -(\gamma^\mu)^T$. (Peter van Nieuwenhuizen calls $\psi^T C$ the Majorana adjoint, and his discussion seems simpler than P&S.)
In 4 Euclidean dimensions the expression $C (\gamma^\mu \partial_\mu +m)$ is skew symmetric in the combined spin and $x$ space, so just as the "variation" of
the quadratic Grassmann expression
$$
S={\textstyle \frac 12} \xi^TA\xi
$$
with $A$ a skew-symmetric matrix gives $A\xi=0$,
the variation of gives the equation of motion
$$
(\gamma^\mu \partial_\mu +m)\psi=0.
$$
In Minkowski signature the four-component spinor is no longer a Grassman variable but is operator-valued with $\bar\psi= \psi^\dagger \gamma^0$ with $\dagger$ implying the adjoint operation on the many-body Hilbert space as usual. The field $\psi$ however, satisfies the Majorana condition
$$
\psi=\psi^c\equiv C^{-1} \bar\psi^T
$$
and the anticommutation commutation relations
$$
\{\psi_\alpha(x), \psi_\beta(x')\}_{t=t'} = [\gamma^0 C^{-1}]_{\alpha\beta} \delta^3(x-x').
$$
In the space-time dimensions ($d=2,3,4$ mod 8) where Majorana's exist the matrix $\gamma^0 C^{-1}$ is symmetric --- as it needs to be for the anticommutator to be consistent with the RHS.
I think that in the $\gamma^5$ diagonal representation of the Gamma matrices we have
$$
C= \left[\matrix{0& i \sigma_2\\ -i\sigma_2&0}\right],
$$
so it should be possible to match my basis-independent notation to that of P&S.
