  Primordial black holes (PBHs) may form from the collapse of matter
overdensities shortly after the Big Bang. One may identify their existence by
observing gravitational wave (GW) emissions from merging PBH binaries at high
redshifts $z\gtrsim 30$, where astrophysical binary black holes (BBHs) are
unlikely to merge. The next-generation ground-based GW detectors, Cosmic
Explorer and Einstein Telescope, will be able to observe BBHs with total masses
of $\mathcal{O}(10-100)~M_{\odot}$ at such redshifts. This paper serves as a
companion paper of arXiv:2108.07276, focusing on the effect of higher-order
modes (HoMs) in the waveform modeling, which may be detectable for these high
redshift BBHs, on the estimation of source parameters. We perform Bayesian
parameter estimation to obtain the measurement uncertainties with and without
HoM modeling in the waveform for sources with different total masses, mass
ratios, orbital inclinations and redshifts observed by a network of
next-generation GW detectors. We show that including HoMs in the waveform model
reduces the uncertainties of redshifts and masses by up to a factor of two,
depending on the exact source parameters. We then discuss the implications for
identifying PBHs with the improved single-event measurements, and expand the
investigation of the model dependence of the relative abundance between the BBH
mergers originating from the first stars and the primordial BBH mergers as
shown in arXiv:2108.07276.
