You have an open modern clay type of chimney flue. The flue from the stove more than likely proceeds past the register plate and then vents into your clay flue. Water being driven or blown down the chimney will therefore fall onto the register plate, with the exception of the few unlucky drops that will fall into the metal flue coming out of the stove.
This is why you have water collecting on your register plate. Even with a rain cowl fitted to your chimney pot, the horizontal rain will compromise this.
The concern has to be that you are using the stove regularly, which in itself should evaporate any water that collects on the register plate.
Is the cowl is still on your chimney pot? The recent extreme weather is showing up all kinds of flaws in original build or poor repairs with ingress of water, damp, mould etc etc, so it may be that water has found its way past a poorly fitted piece of your clay liner?