quantum field theory - 2 Component Spinor Formalism

In Chapters 34-36 of the Srednicki QFT book, 2 component spinors and their combinations in Dirac and Majorana spinors are carefully constructed. Specifically, in equations 36.14 and 36.15 the following left-handed spinors are defined:

$$\begin{equation} \begin{split} &\chi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2)\\ &\xi = \frac{1}{\sqrt{2}}(\psi_1 - i\psi_2) \end{split} \end{equation}$$

A couple of pages later, charge conjugation is then defined as "Charge conjugation simply exchanges $\chi$ and $\xi$".

Now, I have read in other books (e.g. Georgi) that if a particle is defined in one SU(2) representation of the Lorentz group (e.g. as left handed spinor), then the corresponding anti-particle is in the other representation (e.g. as right handed spinor).However, in the above definition in Srednicki, both fermion and anti-fermion are left-handed spinors.

Have I got the this right? How can both of these statements be correct? Any help to clear up my confusion would be greatly appreciated.

As an aside, I have a more general issue understanding the intuition behind Dirac spinors. The state of a general fermion or anti-fermion is often given with a Dirac spinor $\Psi$. If this fermion has mass, then the fermion is a combination of a left-handed and right handed spinor, as in Srednicki:

$$\Psi = \begin{pmatrix} \chi_c \\ \xi^{\dagger\dot{c}} \end{pmatrix}.$$

As helicity and handedness are frame depenendent, I can kind of understand how the Dirac fermion can be a combination of the two above. However, Dirac fermions can also by viewed as a combination of a fermion and anti-fermion in much the same way. This I do not fully understand. How can a fermion with mass be described by a field that is part particle and part anti-particle?

Again, any insights that the community could provide would be much appreciated.