  Molecular based spin-1/2 triangular lattice systems such as
LiZn$_2$Mo$_3$O$_8$ have attracted research interest. Distortions, defects, and
intersite disorder are suppressed in such molecular-based magnets, and
intrinsic geometrical frustration gives rise to unconventional and unexpected
ground states. Li$_2$$A$Mo$_3$O$_8$ ($A$ = In or Sc) is such a compound where
spin-1/2 Mo$_3$O$_{13}$ clusters in place of Mo ions form the uniform
triangular lattice. Their ground states are different according to the $A$
site. Li$_2$InMo$_3$O$_8$ undergoes conventional $120^\circ$ long-range
magnetic order below $T_\text{N}=12$~K whereas isomorphic Li$_2$ScMo$_3$O$_8$
exhibits no long-range magnetic order down to 0.5~K. Here, we report exotic
magnetisms in Li$_2$InMo$_3$O$_8$ and Li$_2$ScMo$_3$O$_8$ investigated by muon
spin rotation ($\mu$SR) and inelastic neutron scattering (INS) spectroscopies
using polycrystalline samples. Li$_2$InMo$_3$O$_8$ and Li$_2$ScMo$_3$O$_8$ show
completely different behaviors observed in both $\mu$SR and INS measurements,
representing their different ground states. Li$_2$InMo$_3$O$_8$ exhibits spin
wave excitation which is quantitatively described by the nearest neighbor
anisotropic Heisenberg model based on the $120^\circ$ spin structure. In
contrast, Li$_2$ScMo$_3$O$_8$ undergoes short-range magnetic order below 4~K
with quantum-spin-liquid-like magnetic fluctuations down to the base
temperature. Origin of the different ground states is discussed in terms of
anisotropies of crystal structures and magnetic interactions.
