Unlike Arcane Ward, Projected Ward is a reaction triggered by seeing a creature "take damage". Injury Poison exposure is also triggered by a creature taking (certain types of) damage.
The text of Arcane Ward (an ongoing effect) says that the ward "takes the damage instead", but Projected Ward pointedly does not restate that. It's clear that if the ward takes the damage instead, that a spell effect isn't affected by poison.
But for Projected Ward's reaction to trigger, a creature (not a spell effect) must have already taken damage, and creatures who take damage from a poisoned weapon have been exposed to injury poison.
Unless the trigger "takes damage" is to be read as "is about to take damage", how does that work with a reaction? If not, should "absorbing" the damage work like a prepared healing spell triggered by seeing someone take damage?
Honestly, you should just write your own answer here. I doubt that you're going to change Exempt-Medic's opinion, so you should make an answer and let votes fall where they will – Tal Feb 17 '22 at 15:28