  We use an exact numerical diagonalization method to calculate the dynamical
spin structure factors (DSFs) of three ab-initio models and one
ab-initio-guided model for a honeycomb-lattice magnet $\alpha$-RuCl$_3$. We
also use thermal pure quantum states to calculate the temperature dependence of
the heat capacity, the nearest-neighbor (NN) spin-spin correlation function,
and the static spin structure factor. From the results obtained from these four
effective models, we find that, even when the magnetic order is stabilized at
low temperature, the intensity at the $\Gamma$ point in the DSFs increases with
increasing NN spin correlation. In addition, we find that the four models fail
to explain heat-capacity measurements whereas two of the four models succeed in
explaining inelastic-neutron-scattering (INS) experiments. In the four models,
when temperature decreases, the heat capacity shows a prominent peak at a high
temperature where the NN spin-spin correlation function increases. However, the
peak temperature in heat capacity is too low in comparison with that observed
experimentally. To address these discrepancies, we propose an effective model
that includes strong ferromagnetic Kitaev coupling, and we show that this model
quantitatively reproduces both INS experiments and heat-capacity measurements.
To further examine the adequacy of the proposed model, we calculate the field
dependence of the polarized terahertz spectra, which reproduces the
experimental results: the spin-gapped excitation survives up to an onset field
where the magnetic order disappears and the response in the high-field region
is almost linear. Based on these numerical results, we argue that the
low-energy magnetic excitation in $\alpha$-RuCl$_3$ is mainly characterized by
interactions such as off-diagonal interactions and weak Heisenberg interactions
between NN pairs, rather than by the strong Kitaev interactions.
