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)Comment: 17 pages, 19 figures, accepted for publication in MNRA