Displacement current: $i_d=\varepsilon_0\dfrac{d\Phi_E}{dt}$ — an effective current due to a time-varying electric flux, introduced by Maxwell to keep Ampere's circuital law consistent (continuity of current) even in regions with no conduction current, such as the gap between capacitor plates.

For a charging capacitor: field between plates $E=\dfrac{Q}{\varepsilon_0A}$, so flux $\Phi_E=EA=\dfrac{Q}{\varepsilon_0}$.

$$i_d=\varepsilon_0\frac{d\Phi_E}{dt}=\varepsilon_0\times\frac{1}{\varepsilon_0}\frac{dQ}{dt}=\frac{dQ}{dt}=i_C$$

So $i_d$ equals the conduction current $i_C$ charging the capacitor at that instant.

For a conductor with a CONSTANT applied voltage: the current is steady (dc), so the electric field inside/flux does not change with time, $\dfrac{d\Phi_E}{dt}=0$, hence $i_d=0$.