Are you only noticing damp under the wooden sill at the opening side of the door (as per 1st image) or are you noticing damp further along i.e. towards the hinge side?
Has it always been doing this or only started recently?
Does it happen after any type of rainfall, or only after heavy/driving rain?
Do you notice any dampness above the wooden sill, or only below?
Can you take a pic at ground level on the outside, looking directly towards the white and grey sills? If the damp is only appearing below the wooden sill, and if it only happens after heavy/driving rain, part of me is wondering if water is getting in between those sills on the outside.
Or is water maybe getting in at the gap in the cladding and tracking down.
You can also try little experiments. when it's had a day or two to dry out, use a hose/watering can to put some water over specific areas. e.g. try putting water over the opening side down near the sills. Wait a little while and note if damp on interior starts to show. If not try other areas. If this doesn't recreate the damp, if possible try forcing a little water between those white and grey sills to determine if that's the problem area.
I'd try all of this prior to taking the internal wooden sill off. However if you're not getting anywhere and if the wooden sill is easy enough to remove, that would possibly help in terms of seeing what's going on.
Also remember, although silicon can look ok, if any sections have slightly failed, that can also be a route in. Check the silicon is still adhered to all surfaces.