The complete reading of the rule about resistance and vulnerability reads as (emphasis mine):
Resistance and then vulnerability are applied after all other modifiers to damage. For example, a creature has resistance to bludgeoning damage and is hit by an attack that deals 25 bludgeoning damage. The creature is also within a magical aura that reduces all damage by 5. The 25 damage is first reduced by 5 and then halved, so the creature takes 10 damage.
The Arcane Ward falls exactly in the example provided in the ruling itself: the damage is firstly reduced by the Ward, without considering any resistance and/or vulnerability that the wizard may have, then immunity, resistance and/or eventually vulnerability effects can be applied to the received damage which affects the wizard.
This is confirmed by the Sage Advice Compendium, which gives the exact order for computing the final damage, including all the relevant characteristics such as immunities, resistances, vulnerabilities and temporary HPs:
How does Arcane Ward interact with temporary hit
points and damage resistance that an abjurer might
have? An Arcane Ward is not an extension of the wizard who creates it. It is a magical effect with its own hit points. Any temporary hit points, immunities, or resistances that the wizard has don’t apply to the ward.
The ward takes damage first. Any leftover damage is taken by the wizard and goes through the following game elements in order: (1) any relevant damage immunity, (2) any relevant damage resistance, (3) any temporary hit points, and (4) real hit points.