  We present the cross section for $e^{+}e^{-}\to hZ$ with arbitrary sets of
electron and $Z$ boson polarizations at the full next-to-leading order in
various extended Higgs models, such as the Higgs singlet model (HSM), the inert
doublet model (IDM) and the two Higgs doublet model (2HDM). We systematically
perform complete one-loop calculations to the helicity amplitudes in the
on-shell renormalization scheme, and present the full analytic results as well
as numerical evaluations. The deviation $\Delta R^{hZ}$ in the total cross
section from its standard model (SM) prediction is comprehensively analyzed,
and the differences among these models are discussed in details. We find that
new physics effects appearing in the renormalized $hZZ$ vertex almost govern
the behavior of $\Delta R^{hZ}$, and it takes a negative value in most cases.
The possible size of $\Delta R^{hZ}$ reaches several percent under the
theoretical and experimental bounds. We also analyze the deviation $\Delta
R^{hZ}_{XY}$ in the total cross section times decay branching ratios of the
discovered Higgs boson by utilizing the $\texttt{H-COUP}$ program. It is found
that the four types of 2HDMs can be discriminated by analyzing the correlation
between $\Delta R^{hZ}_{\tau\tau}$ and $\Delta R^{hZ}_{bb}$ and those between
$\Delta R^{hZ}_{\tau\tau}$ and $\Delta R^{hZ}_{cc}$. Furthermore, the HSM and
the IDM can be discriminated from the 2HDMs by measuring $\Delta R^{hZ}_{WW}$.
These signatures can be tested by precision measurements at future Higgs
factories such as the International Linear Collider.
