  We extend the analysis of Pace et al., JCAP, 2019, 060, by considering the
virialization process in the extended spherical collapse model for clustering
dark-energy models, i.e., accounting for dark-energy fluctuations. Differently
from the standard approach, here virialization is naturally achieved by
properly modelling deviations from sphericity due to shear and rotation induced
by tidal interactions. We investigate the time evolution of the virial
overdensity $\Delta_\mathrm{vir}$ in seven clustering dynamical dark energy
models and compare the results to the $\Lambda$CDM model and to the
corresponding smooth dark-energy models. Taking into account all the
appropriate corrections, we deduce the abundance of convergence peaks for Rubin
Observatory-LSST and Euclid-like weak-lensing surveys, of Sunyaev-Zel'dovich
peaks for a Simon Observatory-like CMB survey, and of X-ray peaks for an
eROSITA-like survey. Despite the tiny differences in $\Delta_\mathrm{vir}$
between clustering and smooth dark-energy models, owing to the large volumes
covered by these surveys, five out of seven clustering dark-energy models can
be statistically distinguished from $\Lambda$CDM. The contribution of
dark-energy fluctuation cannot be neglected, especially for the
Chevallier-Polarski-Limber and Albrecht-Skordis models, provided the
instrumental configurations provide high signal-to-noise ratio. These results
are almost independent of the tidal virialization model.
