  Transition metal dichalcogenides (TMDs) in the $1T$ polymorph are subject to
a rich variety of periodic lattice distortions, often referred to as charge
density waves (CDW) when not too strong. We study from first principles the
fermiology and phonon dispersion of three representative single-layer
transition metal disulfides with different occupation of the $t_{2g}$ subshell:
TaS$_2$ ($t_{2g}^1$), WS$_2$ ($t_{2g}^2$), and ReS$_2$ ($t_{2g}^3$) across a
broad range of doping levels. While strong electron-phonon interactions are at
the heart of these instabilities, we argue that away from half-filling of the
$t_{2g}$ subshell, the doping dependence of the calculated CDW wave vector can
be explained from simple fermiology arguments, so that a weak-coupling nesting
picture is a useful starting point for understanding. On the other hand, when
the $t_{2g}$ subshell is closer to half-filling, we show that nesting is
irrelevant, while a real-space strong-coupling picture of bonding Wannier
functions is more appropriate and simple bond-counting arguments apply. Our
study thus provides a unifying picture of lattice distortions in $1T$ TMDs that
bridges the two regimes, while the crossover between these regimes can be
attained by tuning the filling of the $t_{2g}$ orbitals.
