  Motivated by recent experimental suggestions of charge-order-driven
ferroelectricity in organic charge-transfer salts, such as
$\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Cl, we investigate magnetic and
charge-ordered phases that emerge in an extended two-orbital Hubbard model on
the anisotropic triangular lattice at $3/4$ filling. This model takes into
account the presence of two organic BEDT-TTF molecules, which form a dimer on
each site of the lattice, and includes short-range intramolecular and
intermolecular interactions and hoppings. By using variational wave functions
and quantum Monte Carlo techniques, we find two polar states with charge
disproportionation inside the dimer, hinting to ferroelectricity. These
charge-ordered insulating phases are stabilized in the strongly correlated
limit and their actual charge pattern is determined by the relative strength of
intradimer to interdimer couplings. Our results suggest that ferroelectricity
is not driven by magnetism, since these polar phases can be stabilized also
without antiferromagnetic order and provide a possible microscopic explanation
of the experimental observations. In addition, a conventional dimer-Mott state
(with uniform density and antiferromagnetic order) and a nonpolar
charge-ordered state (with charge-rich and charge-poor dimers forming a
checkerboard pattern) can be stabilized in the strong-coupling regime. Finally,
when electron-electron interactions are weak, metallic states appear, with
either uniform charge distribution or a peculiar $12$-site periodicity that
generates honeycomb-like charge order.
