quantum field theory - Is Parity really violated? (Even though neutrinos are massive)

The weak force couples only to left-chiral fields, which is expressed mathematically by a chiral projection operator $P_L = \frac{1-\gamma_5}{2}$ in the corresponding coupling terms in the Lagrangian.

This curious fact of nature is commonly called parity violation and I'm wondering why? Does this name make sense from a modern point of view?

My question is based on the observation:

A Dirac spinor (in the chiral representation) of pure chirality transforms under parity transformations:

$$ \Psi_L = P_L \Psi = \begin{pmatrix} \chi_L \\ 0 \end{pmatrix} \rightarrow \Psi_L^P = \begin{pmatrix} 0\\ \chi_L \end{pmatrix} \neq \Psi_R$$

Chirality is a Lorentz invariant quantity and a left-chiral particle is not transformed into a right-chiral particle by parity transformations.(The transformed object lives in a different representation of the Lorentz group, where the lower Weyl spinor denotes the left-chiral part.)

In understand where the name comes from historically (see the last paragraph) but wouldn't from a modern point of view chirality violation make much more sense?

Some background:

Fermions are described by Dirac spinors, transforming according to the $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ representation of the (double cover of the) Lorentz group. Weyl spinors $\chi_L$ transforming according to the $(\frac{1}{2},0) $ representation are called left-chiral and those transforming according to the $(0,\frac{1}{2})$ representation are called right-chiral $\xi_R$. A Dirac spinor is (in the chiral representation)

$$ \Psi = \begin{pmatrix} \chi_L \\ \xi_R \end{pmatrix}$$

The effect of a parity transformation is

$$ (\frac{1}{2},0) \underbrace{\leftrightarrow}_P (0,\frac{1}{2}),$$ which means the two irreps of the Lorentz group are exchanged. (This can be seen for example by acting with a parity transformation on the generators of the Lorentz group). That means a parity transformed Dirac spinor, transforms according to the $(0,\frac{1}{2}) \oplus (\frac{1}{2},0) $ representation, which means we have

$$ \Psi = \begin{pmatrix} \chi_L \\ \xi_R \end{pmatrix} \rightarrow \Psi^P = \begin{pmatrix} \xi_R \\ \chi_L \end{pmatrix} $$

Now we can examine the effect of a parity transformation on a state with pure chirality:

$$ \Psi_L = P_L \Psi = \begin{pmatrix} \chi_L \\ 0 \end{pmatrix} \rightarrow \Psi_L^P = \begin{pmatrix} 0\\ \chi_L \end{pmatrix}$$

This means we still have a left-chiral spinor, only written differently, after a parity transformation and not a right-chiral. Chirality is a Lorentz invariant quantity. Nevertheless, the fact that only left-chiral particles interact weakly is commonly called parity violation and I'm wondering if this is still a sensible name or only of historic significance?

Short remark on history

I know that historically neutrinos were assumed to be massless, and for massless particles helicity and chirality are the same. A parity transformation transforms a left-handed particle into a right-handed particle. In the famous Wu experiment, only left-handed neutrinos could be observed, which is were the name parity violation comes from. But does this name make sense today that we know that neutrinos have mass, and therefore chirality $\neq$ helicity.