physical chemistry - Distinguish between two possible configurations for angular momentum in carbon atom

Consider the possible values of $S$ and $L$ for carbon configuration $1s^2 2s^2 2p^2$ and the corresponding rapresentations with arrows indicating the spins (consider only $S=0,L=0$ and $S=0,L=2$, as for $S=1,L=1$ some are missing).

I can't understand which one of the two representations indicated in red belong to $S=0,L=0$ state and which one to the $S=0,L=2$ state (I took a guess making the picture).

The spin wavefunction is in all cases antysymmetric ($S=0$) and equal to

$$\chi =\left(\, |\!\uparrow_1 \downarrow_2\rangle-|\!\downarrow_1 \uparrow_2\rangle\,\right)\div\sqrt 2$$

Therefore the spatial wavefunction $(L)$ must be symmetric for Pauli principle and in the second red configuration is

$$\Psi =p_0 (1) p_0 (2)$$ in the first one is

$$\Psi= \left[p_- (1) p_+ (2)+p_+ (1) p_- (2)\right]\div\sqrt2$$

The wavefunctions are different so there must be a difference between the two red configurations, but which of them should belong to $S=0,L=2$ and the other to $S=0,L=0$ and why?