Aims. This paper aims to revisit the kinematical and physical properties of the warm (T ~ 5000–10 000 K) atomic gas in the inner disk (<5 au) region of classical T Tauri stars (CTTs) and relate them to the properties of the outer dusty disk resolved with ALMA. We also want to define constraints for the mass-loss in the inner atomic winds and jets to assess their role in the evolution and dispersal of planet-forming disks. Methods. We used the high resolution (R = 115 000, ~2.6 km s−1) spectra of 36 CTTs observed as part of the GIARPS High-resolution Observations of T Tauri stars (GHOsT) project and analysed the profile and luminosity of the brightest optical forbidden lines, namely [O I] 630 nm, [O I] 557 nm, [S II] 406 nm, [S II] 673 nm, and [N II] 658 nm. Results. We decomposed the line profiles into different velocity components, and concentrated our analysis mostly on the so-called narrow low-velocity component (NLVC). We find that about 40% of sources display a NLVC peak velocity (Vp) compatible with the stellar velocity. These include the transitional disks (TD) and typically show a single low velocity component (LVC), lower mass accretion rates, and the absence of a jet. They therefore might represent later evolutionary stages where the emission from the disk is dominant with respect to the wind contribution. No difference in kinematical properties was instead found between sources with full disks and disks with substructures as resolved by ALMA. The [O I] 630 nm profiles peaking at the stellar velocity are well fitted by a simple Keplerian disk model, where the emission line region extends from ~0.01 au up to several tens of au in some cases. The [O I] emission is detected inside the sub-millimetre dust cavities of all the TDs. No correlation is found between Rkep, derived from the line half width at half maximum (HWHM), and the size of the dust cavity. We see an anti-correlation between the [O I] 557/630 nm ratio and Rkep, which suggests that the [O I] emitting region expands as the gas dominating the emission cools and becomes less dense. We confirmed previous findings that the line ratios observed in the LVC, if compared with a thermal single temperature and density model, imply ne ~ 106–108 cm−3 and Te ~ 5000–10 000 K, and additionally constrained the ionisation fraction in the NLVC to be xe < 0.1. We however discuss the limits of applying this diagnostic to winds that are not spatially resolved. Conclusions. The emission from the disk should be considered as an important contribution to the forbidden line emission in CTTs. Also, the clearing of warm atomic gas from the upper disk layers does not seem to follow the dispersal of the bulk of molecular gas and dust during late disk evolution. For the outflow component, we estimated the mass-loss for both the disk winds and jets. We conclude that without better knowledge of the wind geometry and spatial extent, and given the limitation of the diagnostics, the mass-loss rates in the wind traced by the blue shifted LVC cannot be constrained better than a factor of 100, with a Ṁwind/Ṁacc spanning between ~0.01 and more than 1. When compared with synthetic [O I] 630 nm images of X-ray photoevaporation models, the estimated Ṁwind represents a lower limit to the total mass-loss rate of the model, indicating that [O I] 630 nm is likely not the best tracer to probe mass-loss in low-velocity winds.
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