  We extend here a previous investigation on the characteristics of Galactic
carbon stars using more accurate EDR3 astrometry measurements. Based on a much
larger statistics, we confirm that N- and SC-type carbon stars share a very
similar luminosity function, while the luminosities of J-type stars (Mbol) are
fainter by half a magnitude on average. R-hot type carbon stars have
luminosities throughout the RGB, which favours the hypothesis of an external
origin for their carbon enhancement. Moreover, the kinematic properties of a
significant fraction of the R-hot stars are compatible with the thick-disc
population, in contrast with that of N- and SC-type stars, which would belong
mostly to the thin disk. We also derive the luminosity function of a large
number of Galactic extrinsic and intrinsic (O-rich) S stars and show that the
luminosities of the latter are typically higher than the predicted onset of the
third dredge-up during the AGB for solar metallicity. This result is consistent
with these stars being genuine thermally pulsing AGB stars. On the other hand,
using the so-called Gaia-2MASS diagram, we show that the overwhelming majority
of the carbon stars identified in the LAMOST survey as AGB stars are probably
R-hot and/or CH-type stars. Finally, we report the identification of 2660 new
carbon stars candidates that we identified through their 2MASS photometry,
their Gaia astrometry, and their location in the Gaia-2MASS diagram.
