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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° long-range magnetic order below
T
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 (
μ
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
μ
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° 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.