The H3+ molecule has been detected in many lines of sight within the central molecular zone (CMZ) with exceptionally large column densities and unusual excitation properties compared to diffuse local clouds. The detection of the (3,3) metastable level has been suggested to be the signature of warm and diffuse gas in the CMZ. We use the Meudon PDR code to re-examine the relationship between the column density of H3+ and the cosmic-ray ionization rate, $\zeta$, up to large values of $\zeta$. We study the impact of the various mechanisms that can excite H3+ in its metastable state. We produce grids of PDR models exploring different parameters ($\zeta$, size of clouds, metallicity) and infer the physical conditions that best match the observations toward ten lines of sight in the CMZ. For one of them, Herschel observations of HF, OH+, H2O+, and H3O+ can be used as additional constraints. We check that the results found for H3+ also account for the observations of these molecules. We find that the linear relationship between N(H3+) and $\zeta$ only holds up to a certain value of the cosmic-ray ionization rate, which depends on the proton density. A value $\zeta \sim 1 - 11 \times 10^{-14}$ s$^{-1}$ explains both the large observed H3+ column density and its excitation in the metastable level (3,3) in the CMZ. It also reproduces N(OH+), N(H2O+) and N(H3O+) detected toward Sgr B2(N). We confirm that the CMZ probed by H3+ is diffuse, nH $\lesssim$ 100 cm-3 and warm, T $\sim$ 212-505 K. This warm medium is due to cosmic-ray heating. We also find that the diffuse component probed by H3+ must fill a large fraction of the CMZ. Finally, we suggest the warm gas in the CMZ enables efficient H2 formation via chemisorption sites as in PDRs. This contributes to enhance the abundance of H3+ in this high cosmic-ray flux environment.