  We construct the finite-temperature dynamical phase diagram of the fully
connected transverse-field Ising model from the vantage point of two disparate
concepts of dynamical criticality. An analytical derivation of the classical
dynamics and exact diagonalization simulations are used to study the dynamics
after a quantum quench in the system prepared in a thermal equilibrium state.
The different dynamical phases characterized by the type of non-analyticities
that emerge in an appropriately defined Loschmidt-echo return rate directly
correspond to the dynamical phases determined by the spontaneous breaking of
$\mathbb{Z}_2$ symmetry in the long-time steady state. The dynamical phase
diagram is qualitatively different depending on whether the initial thermal
state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical
phase diagram that can be directly related to its equilibrium counterpart, the
latter gives rise to a divergent dynamical critical temperature at vanishing
final transverse-field strength.
