  In this paper, we test the self-consistencies of the standard and the
covariant light-front quark model and study the zero-mode issue via the decay
constants of pseudoscalar ($P$), vector ($V$) and axial-vector ($A$) mesons, as
well as the $P\to P$ weak transition form factors. With the traditional type-I
correspondence between the manifestly covariant and the light-front approach,
the resulting $f_{V}$ as well as $f_{^1\!A}$ and $f_{^3\!A}$ obtained with the
$\lbd=0$ and $\lbd=\pm$ polarization states are different from each other,
which presents a challenge to the self-consistency of the covariant light-front
quark model. However, such a self-consistency problem can be "resolved" within
the type-II scheme, which requires an additional replacement $M\to M_0$
relative to the type-I case. Moreover, the replacement $M\to M_0$ is also
essential for the self-consistency of the standard light-front quark model. In
the type-II scheme, the valence contributions to the physical
quantities~(${\cal Q}$) considered in this paper are alway the same as that
obtained in the standard light-front quark model, $[{\cal Q}]_{\rm val.}=[{\cal
Q}]_{\rm SLF}$, and the zero-mode contributions to $f_{V,^1\!A,^3\!A}$ and
$f_-(q^2)$ exist only formally but vanish numerically, which implies further
that $[{\cal Q}]_{\rm val.}\dot{=} [{\cal Q}]_{\rm full}$. In addition, the
manifest covariance of the covariant light-front quark model is violated in the
traditional type-I scheme, but can be recovered by taking the type-II scheme.
