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Constraining the spatial and thermal structure of the gaseous component of circumstellar disks is crucial for understanding star and planet formation. Models predict that the [Ne II] line at 12.81μm detected in young stellar objects (YSOs) with Spitzer traces disk gas and its response to high-energy radiation, but such [Ne II] emission may also originate in shocks within powerful outflows. To distinguish between these potential origins for mid-infrared [Ne II] emission and to constrain disk models, we observed 32 YSOs using the high-resolution (R ~ 30,000) mid-infrared spectrograph VISIR at the Very Large Telescope. We detected the 12.81μm [Ne II] line in 12 objects, tripling the number of detections of this line in YSOs with high spatial and spectral resolution spectrographs. We obtain the following main results. (1) In Class I objects the [Ne II] emission observed from Spitzer is mainly due to gas at a distance of more than 20-40 AU from the star, where neon is, most likely, ionized by shocks due to protostellar outflows. (2) In transition and pre-transition disks, most of the emission is confined to the inner disk, within 20-40 AU from the central star. (3) Detailed analysis of line profiles indicates that, in transition and pre-transition disks, the line is slightly blueshifted (2-12 km s-1) with respect to the stellar velocity, and the line width is directly correlated with the disk inclination, as expected if the emission is due to a disk wind. (4) Models of EUV/X-ray-irradiated disks reproduce well the observed relation between the line width and the disk inclination, but underestimate the blueshift of the line. [ABSTRACT FROM AUTHOR]