For parallel propagation, use the undivided dispersion relation:

P (n^{2} - R) (n^{2} - L) = 0.

The five requested cases are:

Plasma oscillations: $P = 0$ and $δ E ∥ B_{0} ∥ k$ . With negligible ion response, $ω^{2} = ω_{p e}^{2}$ , independently of $k$ in this cold approximation.
Cold-plasma right-handed waves: $n^{2} = R$ , $E_{y} = i E_{x}$ , $E_{z} = 0$ , for propagation along $+ B_{0}$ . At a fixed point, the real electric field rotates from $+ \hat{x}$ towards $+ \hat{y}$ , in the sense of electron gyration.
Cold-plasma left-handed waves: $n^{2} = L$ , $E_{y} = - i E_{x}$ , $E_{z} = 0$ . The real electric field rotates in the opposite sense.
Ordinary plasma waves: for perpendicular propagation, $θ = π /2$ , $n^{2} = P$ , and $δ E ∥ B_{0}$ . Their dispersion relation is $ω^{2} = ω_{p e}^{2} + c^{2} k^{2}$ .
Extraordinary plasma waves: for perpendicular propagation, $n^{2} = R L / S$ , $E_{z} = 0$ , and $E_{y} = i g E_{x} / (S - n^{2})$ . In general the electric field has both longitudinal and transverse components.