Probing planet formation and disk substructures in the inner disk of Herbig Ae stars with CO rovibrational emission

[abridged]CO rovibrational lines are efficient probes of warm molecular gas and can give unique insights into the inner 10 AU of proto-planetary disks. Recent studies have found a relation between the ratio of lines originating from the second and first vibrationally excited state, denoted as $v2/v1$, and the emitting radius of CO. In disks around Herbig Ae stars the vibrational excitation is low when CO lines come from close to the star, and high when lines only probe gas at large radii (more than 5 AU). We aim to find explanations for the observed trends between CO vibrational ratio, emitting radii, and NIR excess, and identify their implications in terms of the physical and chemical structure of inner disks around Herbig stars. Slab models and full disk thermo chemical models are calculated. Simulated observations from the models are directly compared to the data. Broad CO lines with low vibrational ratios are best explained by a warm (400-1300 K) inner disk surface with gas-to-dust ratios below 1000; no CO is detected within/at the inner dust rim, due to dissociation at high temperatures. In contrast, explaining the narrow lines with high vibrational ratios requires an inner cavity of a least 5 AU in both dust and gas, followed by a cool (100-300 K) molecular gas reservoir with gas-to-dust ratios greater than 10000 at the cavity wall. In all cases the CO gas must be close to thermalization with the dust. The high gas-to-dust ratios needed to explain high $v2/v1$ in narrow CO lines for a subset of group I disks can naturally be interpreted as due to the dust traps that have been proposed to explain millimeter dust cavities. The broad lines seen in most group II objects indicate a very flat disk in addition to inner disk substructures within 10 AU that can be related to the substructures recently observed with ALMA.