We re-address the classification criterion for extremely red galaxies (ERGs) of Pozzetti and Mannucci (2000 -- PM00), which aims to separate, in the Ic-K (or Rc-K) vs. J-K colour--colour diagram, passively evolving, old (> 1 Gyr) stellar populations in a dust-free environment, associated with ellipticals (Es), from dusty starburst galaxies (DSGs), both at 1 < z < 2. We explore a category of objects not considered previously, i.e., galaxies forming in this redshift range on short (0.1 Gyr) timescales and observed also in their early, dusty post-starburst phase. We also investigate the impact of structure of the dusty medium and dust amount on the observed optical/near-IR colours of high-z DSGs/DPSGs, through multiple-scattering radiative transfer calculations for a dust/stars configuration and an extinction function calibrated with nearby dusty starbursts. As a main result, we find that dusty post-starburst galaxies (DPSGs), with ages between 0.2 and 1 Gyr, at 1.3 < z < 2 mix with Es at 1 < z < 2 for a large range in dust amount. This ``intrusion is a source of concern for the present two-colour classification of ERGs. On the other hand, we confirm, in agreement with PM00, that DSGs are well separated from Es, both at 1 < z < 2, in the Ic-K vs. J-K colour--colour diagram, whatever the structure (two-phase clumpy or homogeneous) of their dusty medium and their dust amount are. This result holds under the new hypothesis of high-z Es being as dusty as nearby ones. Thus the interpretation of the optical/near-IR colours of high-z Es may suffer from a multiple degeneracy among age, metallicity, dust and redshift. We also find that DPSGs at z around 1 mix with DSGs at 1 < z < 2, as a function of dust amount and structure of the dusty medium. All these results help explaining the complexity of the ERG classification... (Abridged)
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