The copper-based chalcogenide Cu${}_4$TiSe${}_4$, a material with intrinsically low lattice thermal conductivity, is promising for applications in thermal barrier coatings and thermoelectric conversion. Both its rhombohedral and cubic phases were synthesized and are stable at room temperature and ambient pressure. The only difference in their synthesis is the use of ball milling, which induces a phase transition from the former to the latter. In situ high-pressure X-ray diffraction shows that under hydrostatic pressure this transition occurs at a critical pressure of approximately 4.4 GPa and is reversible upon pressure release. In contrast, the transition induced by the dynamic pressure generated during ball milling is irreversible due to kinetic trapping. Counterintuitively, the cubic phase, despite its higher symmetry, exhibits consistently lower lattice thermal conductivity than the rhombohedral phase across the measured temperature range, with a value about 33% lower at room temperature ( 0.79 vs. 1.18 W m${}^{-1}$ K${}^{-1}$). associated with displacements of Cu2 atoms, which enhances their vibrational dynamics and leads to larger mean square displacements and dynamic disorder in the cubic phase. Consequently, phonon group velocities decrease and phonon lifetimes are shortened, intensifying phonon scattering. These findings suggest that materials with shallow potential wells and pronounced atomic vibrations may exhibit intrinsically low lattice thermal conductivity, offering a strategy complementary to conventional phonon engineering.