Does "chiral fermion" mean "massless fermion"?

Question

I don't understand the term "chiral fermion". To my understanding a fermion $\psi$ can be decomposed as the sum of its left and right (chiral) parts: $$\psi=\psi_L+\psi_R=\frac{1}{2}(1+\gamma^5)\psi+\frac{1}{2}(1-\gamma^5)\psi$$ and the trick is that if in the Lagrangian $m=0$ then the two chiral components propagate independently and the symmetry group of the theory doubles.

When one refers to a "chiral fermion" do they mean a $\psi$ that can be decomposed in its chiral parts (in which case, are there cases where this isn't possible and so the term "chiral fermion" is needed)?
Or do they mean that the symmetry group can be written as the product of the two chiral symmetry groups (in which case, why don't we just say "massless fermion")?

(to be clear, when I say "massive" I mean "with a mass term in the Lagrangian")

Answer 1

The answer is no.

For a non-chiral fermion $$ \psi = \begin{pmatrix} \psi_L \\ \psi_R\end{pmatrix} $$ We can associate a Dirac mass term $$ \overline{\psi_L}\psi_R + \overline{\psi_R}\psi_L $$ For a chiral fermion $\psi_L$, we can associate a Majorana mass term $$ \overline{\psi_L^c}\psi_L+\overline{\psi_L}\psi_L^c $$