  Context. Various Be stars exhibit intensity variations of the violet and red
emission peaks in their HI lines observed in emission. This so-called $V/R$
phenomenon is usually explained by the precession of a one-armed spiral density
perturbation in the circumstellar disk. That global-disk oscillation scenario
was confirmed, both observationally and theoretically, in the previous series
of two papers analyzing the Be shell star {\zeta} Tauri. The vertically
averaged (2D) global-disk oscillation model used at the time was able to
reproduce the $V/R$ variations observed in H{\alpha}, as well as the spatially
resolved interferometric data from AMBER/VLTI. Unfortunately, that model failed
to reproduce the $V/R$ phase of Br15 and the amplitude of the polarization
variation, suggesting that the inner disk structure predicted by the model was
incorrect.
  Aims. The first aim of the present paper is to quantify the temporal
variations of the shell-line characteristics of {\zeta} Tauri. The second aim
is to better understand the physics underlying the $V/R$ phenomenon by modeling
the shell-line variations together with the $V/R$ and polarimetric variations.
The third aim is to test a new 2.5D disk oscillation model, which solves the
set of equations that describe the 3D perturbed disk structure but keeps only
the equatorial (i.e., 2D) component of the solution. This approximation was
adopted to allow comparisons with the previous 2D model, and as a first step
toward a future 3D model.
  Results. The new 2.5D formalism improves the agreement with the observed
$V/R$ variations of H{\alpha} and Br15, under the proviso that a large value of
the viscosity parameter, {\alpha} = 0.8, be adopted. Nonetheless, it remains
challenging for the models to reproduce consistently the amplitude and the
average level of the polarization data, whatever formalism is adopted.
