I will explain in details why I stated (in the comments) that some states are inconsistent. The spin-orbit (S.O) coupling correlates the orbital angular momentum $\vec{L}$ and the spin angular momentum $\vec{L}$ in a way that the eigenvalues of these operators are not good quantum numbers. This means naively that the ground-state is not an eigenvector of these operators anymore, but an eigenvector of the total angular momentum $\vec{J}$. This is a non surprising result as the orbit of a charged particle independent of its spin around a heavy charged particle creates has an angular momentum, a magnetic moment and a magnetic field which acts on the spin of the particle : in a more intuitive picture, one has to consider a relative motion of the heavy charge around the light particle, giving in the frame of the latter the superposition of $\vec{L}$ and $\vec{S}$. When someone is used to deal with dipole interactions, there is an intuitive deduction that the orbital magnetic moment derived from $\vec{L}$ and the spin magnetic moment proportional to $\vec{S}$ should be antiparallel in the ground-state : therefore the classical direction of the rotational motion should be chosen depending on the orientation of the spin to the particle. Considering a weak interaction ($^2P$), the S.O. coupling gives the correction :
$$ \Delta H_{S.O.} = \lambda \vec{L}\cdot \vec{S} = \lambda L.S.\cos(\theta)$$
There is no need to use this expression $ \Delta E_{S.O.} = \beta \left[j(j+1)-l(l+1)-s(s+1)\right]$ to derive the third rule : if $\lambda > 0 $, the only stable configuration is $\cos(\theta) < 0$, this means $ \frac{\pi}{2} < \theta < \frac{3\pi}{2}$. Because, $\vec{L}$ is quantized along $z$, $L_z |l,m\rangle = \hbar m_l|l,m\rangle$ with $m_l$ integers, and $S_z |l,m\rangle = \hbar m_s|s,m\rangle$ with $m_s$ half integers, the angle $\theta \approx \frac{9}{10} \pi \neq \pi$ for an antiparallel configuration of $\vec{L}$ and $\vec{S}$ (should be checked). With S.O. coupling, $m_j$ eigenvalues of $\vec{J}$ are good quantum numbers. The previous results shows that $\vec{S}$ and $\vec{L}$ should be antiparallel for a stable configuration. This means that if you consider $\vec{J} = \vec{L} + \vec{S}$
as the total angular momentum, $m_l<0$ should match $m_s>0$ and inversely, the filling is given step-by-step in the table below :
\begin{array}{|l|c|c|c|c|}\hline
m_l & +1 & 0 & -1\\\hline
& \downarrow & & \\\hline
& \downarrow & \downarrow & \\\hline
& \downarrow & \downarrow & \downarrow \\\hline
& \downarrow & \downarrow & \uparrow\downarrow \\\hline
(4) & \downarrow & \uparrow\downarrow & \uparrow\downarrow \\\hline
\end{array}
The final states gives $J=|-1+\frac{1}{2} |= 3/2$ and is the ground state, some of the other states do not satisfy $\Delta E_{SO} <0$ especially if $J=1/2$. The state (3) is also correct only if one starts by $m_l=-1$ :
\begin{array}{|l|c|c|c|c|}\hline
m_l & -1 & 0 & +1\\\hline
(3) & \uparrow & \uparrow\downarrow & \uparrow\downarrow \\\hline
\end{array}
However, the state (1) represent the conventional filling picture, from left to right, where the empty states are on the right, this is the reason why the third rule has the term $J=|L-S|$ for less than half filled and $J=L+S$ for more than half filled shell. Obviously, if you fill the shell from left to right until an half filled shell an then right to left, $J\equiv |L+S |$.
