Some books, such as Maggiore, write that the axial $U(1)$ transformation is the chiral transformation. A Dirac field $\Psi \in (\frac12, \frac12)$ undergoes a chiral transformation in the following manner.
$$ \Psi \mapsto e^{i\beta\gamma_5}\Psi $$
In a chiral basis, we can write $\Psi \equiv \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix}$ where $\psi_L \in (\frac12,0)$ and $\xi_R \in (0,\frac12)$. Then,
$$ \psi_L \mapsto e^{-i\beta}\psi_L \,,\qquad \xi_R \mapsto e^{i\beta}\xi_R \,. $$
Maggiore defines chiral fields to be ones that violate the axial $U(1)$ symmetry.
Whereas books such as Srednicki define chiral fields to be ones that violate parity. A parity transformation does the following.
$$ \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix} \mapsto \begin{pmatrix} \xi_R\\\psi_L \end{pmatrix} $$
Srednicki discusses the axial $U(1)$ symmetry in a different chapter as a global symmetry in consistent gauge theories.
So, which of the definitions of chirality is true? And if they are equivalent, how so?
parity violation
$\Leftrightarrow$axial asymmetry
? In other words, are the two definitions equivalent? $\endgroup$