  We consider a two-band spinless model describing an excitonic insulator (EI)
on the two-dimensional square lattice with anisotropic hopping parameters and
electron-phonon (el-ph) coupling, inspired by the EI candidate Ta$_2$NiSe$_5$.
We systematically study the nature of the collective excitations in the ordered
phase which originates from the interband Coulomb interaction and the el-ph
coupling. When the ordered phase is stabilized only by the Coulomb interaction
(pure EI phase), its collective response exhibits a massless phase mode in
addition to the amplitude mode. We show that in the BEC regime, the signal of
the amplitude mode becomes less prominent and that the anisotropy in the phase
mode velocities is relaxed compared to the model bandstructure. Through
coupling to the lattice, the phase mode acquires a mass and the signal of the
amplitude mode becomes less prominent. Importantly, character of the softening
mode at the boundary between the normal semiconductor phase and the ordered
phase depends on the parameter condition. In particular, we point out that even
for el-ph coupling smaller than the Coulomb interaction the mode that softens
to zero at the boundary can have a phonon character. We also discuss how the
collective modes can be observed in the optical conductivity. Furthermore, we
study the effects of nonlocal interactions on the collective modes and show the
possibility of realizing a coexistence of an in-gap mode and an above-gap mode
split off from the single amplitude mode in the system with the local
interaction only.
