  We investigate two separate notions of dynamical phase transitions in the
two-dimensional nearest-neighbor transverse-field Ising model on a square
lattice using matrix product states and a new \textit{hybrid} infinite
time-evolving block decimation algorithm, where the model is implemented on an
infinitely long cylinder with a finite diameter along which periodic boundary
conditions are employed. Starting in an ordered initial state, our numerical
results suggest that quenches below the dynamical critical point give rise to a
ferromagnetic long-time steady state with the Loschmidt return rate exhibiting
\textit{anomalous} cusps even when the order parameter never crosses zero.
Within the accessible timescales of our numerics, quenches above the dynamical
critical point suggest a paramagnetic long-time steady state with the return
rate exhibiting \textit{regular} cusps connected to zero crossings of the order
parameter. Additionally, our simulations indicate that quenching slightly above
the dynamical critical point leads to a coexistence region where both anomalous
and regular cusps appear in the return rate. Quenches from the disordered phase
further confirm our main conclusions. Our work supports the recent finding that
anomalous cusps arise only when local spin excitations are the energetically
dominant quasiparticles. Our results are accessible in modern Rydberg
experiments.
