  We reexamine the one-dimensional spin-1 $XXZ$ model with on-site uniaxial
single-ion anisotropy as to the appearance and characterization of the
symmetry-protected topological Haldane phase. By means of large-scale
density-matrix renormalization group (DMRG) calculations the central charge can
be determined numerically via the von Neumann entropy, from which the
ground-sate phase diagram of the model can be derived with high precision. The
nontrivial gapped Haldane phase shows up in between the trivial gapped even
Haldane and N{\'e}el phases, appearing at large single-ion and spin--exchange
interaction anisotropies, respectively. We furthermore carve out a
characteristic degeneracy of the lowest entanglement level in the topological
Haldane phase, which is determined using a conventional finite-system DMRG
technique with both periodic and open boundary conditions. Defining the spin
and neutral gaps in analogy to the single-particle and neutral gaps in the
intimately connected extended Bose-Hubbard model, we show that the excitation
gaps in the spin model qualitatively behave just as for the bosonic system. We
finally compute the dynamical spin structure factor in the three different
gapped phases and find significant differences in the intensity maximum which
might be used to distinguish these phases experimentally.
