Meaning of the subscripts $L,R$ for the two component Weyl spinors $\phi_{L,R}$ For a Dirac spinor $\psi$, its chiral projections are $\psi_{L,R}$ are defined as $$\psi_{R,L}=\frac{1}{2}(1\mp\gamma^5)\psi.\tag{1}$$ Acting with the chirality operator $\gamma^5$, we find $$\gamma^5\psi_L=-\psi_L,~~\gamma^5\psi_R=+\psi_R.\tag{2}$$ This is why $\psi_L$ and $\psi_R$ are respectively known as left-handed and right-handed chiral projections of $\psi$. It is to be emphasized that $\psi_L$ and $\psi_R$ are not 2-component spinors; $\psi_L$($\psi_R$) are still 4-component spinors with lower(upper) two entries being zero and upper(lower) two entries being nonzero. Let $$\psi_L=\begin{pmatrix}\chi\\0\end{pmatrix},~~\psi_R=\begin{pmatrix}0\\\zeta\end{pmatrix},\tag{3}$$ where $\chi$ and $\zeta$ are two-component spinors, called Weyl spinors. But sometimes people use a confusing notation, $\phi_L$ for $\chi$ and $\phi_R$ for $\zeta$ i.e., $$\psi_L=\begin{pmatrix}\phi_L\\0\end{pmatrix},~~\psi_R=\begin{pmatrix}0\\\phi_R\end{pmatrix}.\tag{4}$$  For example, see Eq. (8.71) here. 
Since the chirality projection operators $\frac{1}{2}(1\mp\gamma^5)$ are $4\times4$ matrices, they can only act on $\psi$ to project out $\psi_{L}$ and $\psi_R$. However, the notation $\phi_L$ and $\phi_R$, for the 2-component spinors $\chi$ and $\zeta$ respectively, suggests that there is also a notion of $2\times 2$ chirality operator. If there is no such operator what is the meaning of $\phi_L$ and $\phi_R$?   
 A: It's important to distinguish between the Clifford algebra itself versus a matrix representation of the Clifford algebra. The Clifford algebra itself is an abstract associative algebra generated by basis vectors $e^0,e^1,e^2,e^3$ satisfying $e^a e^b+e^be^a=2\eta^{ab}$. The Dirac matrices provide a matrix representation of the Clifford algebra, $\gamma:e^a\mapsto \gamma^a$, which is faithful in the sense that distinct elements of the Clifford algebra are represented by distinct matrices. In four-dimensional spacetime, the smallest matrices that can achieve this feat have size $4\times 4$.
A Dirac spinor is a thing that is acted on by this faithful matrix representation of the whole Clifford algebra.
The even part of the Clifford algebra is generated by products $e^a e^b$. It is a proper subalgebra of the full Clifford algebra. When restricted to this subalgebra, the Dirac matrix representation is reducible: using the projection matrices $(1\pm\gamma^5)/2$, we can split a Dirac spinor $\psi$ into two parts $\psi_{L/R}$ that don't mix with each other under the action of the even part of this representation of the Clifford algebra. 
Without referring to Dirac spinors at all, Weyl spinors (aka chiral spinors) can be defined directly as things that transform according to an irreducible representation of the even part of the Clifford algebra. There are two inequivalent (mutually conjugate) representations of the even part, which are often distinguished from each other using the subscripts $L/R$, whether or not they were constructed by applying $(1\pm\gamma^5)/2$ to a Dirac spinor.
When Weyl spinors are defined directly like this, the chirality operator is still defined: it's still (proportional to) the matrix representation of $e^0e^1e^2e^3$. However, an irreducible representation of the even part of the Clifford algebra is not faithful: the matrix representing $e^0e^1e^2e^3$ is proportional to the identity matrix. The two inequivalent representations differ from each other in the sign of the matrix that represents $e^0e^1e^2e^3$. So the chirality operator is still defined, but it just multiplies the Weyl spinor by $+1$ or $-1$, depending on which of the two inequivalent representations is being used.
