Capacitors: why is the energy not stored in a magnetic field?

When a capacitor is charging, the rate of change $dE/dt$ of the electric field between the plates is non-zero, and from the Maxwell-Ampère equation this causes a circulating magnetic field.

Now, since a magnetic field exists, why is the energy of a capacitor only stored in the electric field? Usually the formula for the energy stored goes as $W = \pi d A \times \frac{1}{2}\epsilon_0 E^2$, where the first term is the volume and latter is the electric field energy density.

In Poynting's theorem, the electro-magnetic field energy density is $\frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$, i.e. there is also the magnetic field B present.

In a capacitor B is non-zero, so why do we not include it in the calculation of the energy stored?

In other words, why is the energy stored in a capacitor just $[ (volume) \times \frac{1}{2}\epsilon_0 E^2 ]$ and not $[ (volume) \times (\frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2) ]$ ?

You are correct, that while charging a capacitor there will be a magnetic field present due to the change in the electric field. And of course $B$ contains energy as pointed out. However: As the capacitor charges, the magnetic field does not remain static. This results in electromagnetic waves which radiate energy away. The energy put into the magnetic field during charging is lost in the sense that it cannot be feed back to the circuit by the capacitor.