I was reading through Lagrangian Mechanics and noticed this sort of question popping up everywhere. To illustrate, the 3d x coordinate was considered to be a function of X, Y (degrees of freedom). Now in the differentiating mess, the authors considered both x and X as functions of time. But I thought x was a function of X and hence the confusion.
If Lagrangian is written as $L=L(x(t),v(t),t)$, $L$ is time-dependent explicity through $t$ and implicitely through $x$ and $v$. If $L=L(x(t),v(t))$, then $L$ is not explicitly dependent on time but implicitly dependent on time through $x$ and $v$. This means $L$ can change if and only if $x$ and $v$ changes with time and can't change with time without a change in $x$ and $v$.
Same explanation goes for all the functions depending explicitly and implicitly on time.