  The topotactic intercalation of transition-metal dichalcogenides with atomic
or molecular ions acts as an efficient knob to tune the electronic ground state
of the host compound. A representative material in this sense is
1$T$-TiSe$_{2}$, where the electric-field-controlled intercalations of lithium
or hydrogen trigger superconductivity coexisting with the charge-density wave
phase. Here, we use the nuclear magnetic moments of the intercalants in
hydrogen-intercalated 1$T$-TiSe$_{2}$ as local probes for nuclear magnetic
resonance experiments. We argue that fluctuating mesoscopic-sized domains
nucleate already at temperatures higher than the bulk critical temperature to
the charge-density wave phase and display cluster-glass-like dynamics in the
MHz range tracked by the $^{1}$H nuclear moments. Additionally, we observe a
well-defined independent dynamical process at lower temperatures that we
associate with the intrinsic properties of the charge-density wave state. In
particular, we ascribe the low-temperature phenomenology to the collective
phason-like motion of the charge-density wave being hindered by structural
defects and chemical impurities and resulting in a localized oscillating
motion.
