Cosmic-rays, gas, and dust in nearby anticentre clouds. II. Interstellar phase transitions and the dark neutral medium

H$_{\rm I}$ 21-cm and $^{12}$CO 2.6-mm line emissions respectively trace the atomic and molecular gas phases, but they miss most of the opaque H$_{\rm I}$ and diffuse H$_2$ present in the Dark Neutral Medium (DNM) at the transition between the H$_{\rm I}$-bright and CO-bright regions. Jointly probing H$_{\rm I}$, CO, and DNM gas, we aim to constrain the threshold of the H$_{\rm I}$-H$_2$ transition. We also aim to measure gas mass fractions in the different phases and to test their relation to cloud properties. We have used dust optical depth measurements at 353 GHz, $\gamma$-ray maps at GeV energies, and H$_{\rm I}$ and CO line data to trace the gas column densities and map the DNM in nearby clouds toward the Galactic anticentre and Chamaeleon regions. The onset of detectable CO intensities varies by only a factor of 4 from cloud to cloud, between 0.6$ \times 10^{21}$ cm$^{-2}$ and 2.5$ \times 10^{21}$ cm$^{-2}$ in total gas column density. We observe larger H$_2$ column densities than linearly inferred from the CO intensities at $A_{\rm{V}}> 3$ magnitudes because of the large CO optical thickness: the additional H$_2$ mass in this regime represents on average 20$\%$ of the CO-inferred molecular mass. In the DNM envelopes, we find that the fraction of diffuse CO-dark H$_2$ in the molecular column densities decreases with increasing $A_{\rm{V}}$ in a cloud. For a half molecular DNM, the fraction decreases from more than 80$\%$ at 0.4 magnitude to less than 20$\%$ beyond 2 magnitudes. In mass, the DNM fraction varies with the cloud properties. The mass present in the DNM envelopes appears to scale with the molecular mass seen in CO as $M_{\rm H}^{\rm DNM} = (62 \pm 7) {M_{\rm H_2}^{\rm CO}}^{\,0.51 \pm 0.02}$ across two decades in mass. The phase transitions in these clouds show both common trends and environmental differences.