  The impact of the streaming between baryons and dark matter on the first
structures has been actively explored by recent studies. We investigate how the
key results are affected by two popular approximations. One is to implement the
streaming by accounting for only the relative motion while assuming ``baryons
trace dark matter" spatially at the initialization of simulation. This neglects
the smoothing on the gas density taking place before the initialization. In our
simulation initialized at $z_i=200$, it overestimates the gas density power
spectrum by up to 40\% at $k\approx10^2~h~\mbox{Mpc}^{-1}$ at $z=20$. Halo mass
($M_h$) and baryonic fraction in halos ($f_{b,h}$) are also overestimated, but
the relation between the two remains unchanged. The other approximation tested
is to artificially amplify the density/velocity fluctuations in the cosmic mean
density to simulate the first minihalos that form in overdense regions. This
gives a head start to the halo growth while the subsequent growth is similar to
that in the mean density. The growth in a true overdense region, on the other
hand, is accelerated gradually in time. For example, raising $\sigma_8$ by 50\%
effectively transforms $z\rightarrow\sqrt{1.5}z$ in the halo mass growth
history while in 2-$\sigma$ overdensity, the growth is accelerated by a
constant in redshift: $z\rightarrow{z+4.8}$. As a result, halos have grown more
in the former than in the latter before $z\approx27$ and vice versa after. The
$f_{b,h}$-$M_h$ relation is unchanged in those cases as well, suggesting that
the Pop III formation rate for a given $M_h$ is insensitive to the tested
approximations.
