Crossover from weak to strong chaos in high-dimensional Hamiltonian systems at the strong stochasticity threshold (SST) was anticipated to indicate a global transition in the geometric structure of phase space. Our recent study of Fermi-Pasta-Ulam models showed that corresponding to this transition the energy density dependence of all Lyapunov exponents is identical apart from a scaling factor. The current investigation of the dynamic XY model discovers an alternative scenario for the energy dependence of the system dynamics at SSTs. Though similar in tendency, the Lyapunov exponents now show individually different energy dependencies except in the near-harmonic regime. Such a finding restricts the use of indices such as the largest Lyapunov exponent and the Ricci curvatures to characterize the global transition in the dynamics of high-dimensional Hamiltonian systems. These observations are consistent with our conjecture that the quasi-isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities. Moreover, numerical simulations demonstrate the existence of hydrodynamical Lyapunov modes (HLMs) in the dynamic XY model and show that corresponding to the crossover in the Lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of HLMs. In particular, our numerical results confirm that strong chaos is essential for the appearance of HLMs.