Dirac vs KG propagation amplitude

Can someone explain to me the physical meaning of $\bar{\psi}=\psi^\dagger\gamma^0$ in the Dirac equation? I understand it is obtained as one of the solutions of Dirac equation and it is used to build the scalar $\bar{\psi}\psi$ but I am not sure I understand its physical meaning. I see that when we calculate propagation amplitudes we use $<0|\psi(x)\bar{\psi(y)}|0>$ and not $<0|\psi(x)\psi(y)|0>$ (as we would do in Klein-Gordon equation), so I guess from this that $\bar{\psi(x)}$ applied on the vacuum state creates a particle at position x, but $\psi(x)$ does exactly the same thing (and I mean both create a particle, not antiparticle) so what is the point of that, then?

The Dirac adjoint is defined in order to construct a Lorentz scalar, as $\psi^\dagger \psi$ does not transform as a scalar unless the representation were unitary, which it is not.
$$\psi \sim b + c^\dagger, \quad \psi^\dagger \sim b^\dagger + c$$
ignoring the integration and other factors. This means that $\psi$ containing $c^\dagger$ creates particles so to speak with the spinor $v(p)$ corresponding to negative frequency solutions and $b^\dagger$ in $\psi^\dagger$ contributes positive frequency solutions $u(p)$ of the Dirac equation.