As of 3.5, traits that trigger when taking damage (orc traits, counterattack, frenzy) also started triggering even if the damage you took only hit armor. However, straight armor reduction still doesn’t count as “damage” with respect to barrier interactions, which is my litmus test to whether something is “damage” or “not damage”, which means these traits are specifically coded to also trigger from armor reduction. However, also in 3.5, having your armor burning will now hit and consume the barrier, whereas before it would only consume barrier if it was going to deal damage to life, meaning they also changed how burning damage was applied. However, reflected skull damage (Infernal Armor, Thorns) that is decidedly “damage” but only happens to hit armor on that one hit still will not interact with barrier, despite still being “damage” dealt to the troop matching skulls - if the damage spills over past armor and into life, then it will consume your remaining armor and consume barrier, leaving you with full life, no armor, and no barrier. I haven’t tested Deathmark trigger yet, but prior to 3.5, it had a similar interaction with barrier - strip all armor, then deal lethal damage (the damage portion of which is blocked by barrier, leaving you with no armor, no barrier, and deathmark still on the troop). I’m betting this hasn’t changed.
Basically, “damage” still refers to a bunch of different things and is wildly inconsistent. In the case of “when I take damage” traits, “damage” means damage or reduction to armor.
Side note - the last thing I want to see here is a “fix” that makes armor destroy start interacting with barrier. This would allow Death Mark to deal lethal activation damage through barrier only by virtue of being a weirdly coded two-step process and removing a potential counter.