I have seen several posts on here indicating that an Abjurer can use upcast spells to grant more HP to their Arcane Ward.
Is that officially how the rule works?

As I read it:
"Whenever you cast an abjuration spell of 1st leveI or higher, the ward regains a number of hit points equal to twice the leveI of the spell."

e.g., I am a 3rd level wizard with 2nd level spell slots and I cast Shield using a 2nd level spell slot.
Does my Arcane Ward regain 2 HP or 4 HP?

My confusion here hinges on the fact that Shield is a 1st level spell.
Does upcasting it turn it into a 2nd level spell, or is it a 1st level spell cast in a 2nd level spell slot?
Are those the same thing?

Oops, never mind. I just answered my own question.

PHB 201: "When a spellcaster casts a spell using a slot that is a higher level than the spell, the spell assumes the higher level for that casting."

So yes, these things are equivalent and that's why the Arcane Ward gains the higher amount in my example scenario.

Also, from RPGbot's abjurer guide: "If you’re looking for an easy way to charge the ward, consider the Eldritch Initiate feat to get Armor of Shadows. Armor of Shadows allows you to cast Mage Armor on yourself at will, allowing you to fully charge the ward whenever you have a few minutes to repeatedly cast Mage Armor in order to charge it."

Great tip, thank you!