  Transition metal dichalcogenide heterobilayers offer attractive opportunities
to realize lattices of interacting bosons with several degrees of freedom. Such
heterobilayers can feature moir\'e patterns that modulate their electronic band
structure, leading to spatial confinement of single interlayer excitons (IXs)
that act as quantum emitters with $C_3$ symmetry. However, the narrow emission
linewidths of the quantum emitters contrast with a broad ensemble IX emission
observed in nominally identical heterobilayers, opening a debate regarding the
origin of IX emission. Here we report the continuous evolution from a few
trapped IXs to an ensemble of IXs with both triplet and singlet spin
configurations in a gate-tunable $2H$-MoSe$_2$/WSe$_2$ heterobilayer. We
observe signatures of dipolar interactions in the IX ensemble regime which,
when combined with magneto-optical spectroscopy, reveal that the narrow
quantum-dot-like and broad ensemble emission originate from IXs trapped in
moir\'e potentials with the same atomic registry. Finally, electron doping
leads to the formation of three different species of localised negative trions
with contrasting spin-valley configurations, among which we observe both
intervalley and intravalley IX trions with spin-triplet optical transitions.
Our results identify the origin of IX emission in MoSe$_2$/WSe$_2$
heterobilayers and highlight the important role of exciton-exciton interactions
and Fermi-level control in these highly tunable quantum materials.
