ZDoom

The problem you are facing here is that the cases that don't work are not triggered by players. This means you need to enable the 'monsters can activate' flag for any line that is damageable and supposed to be triggered by non-player actors.

This cannot simply be disabled because modders may want to have items only the player can trigger.
In case of the barrels, the ENTER script is activated by a player which gets transferred to the barrel, while the OPEN script has no activator, making the barrel the source of the wall's damage.

It seems that Death Specials on lines with health won't activate if the damage was caused by something other than a player or a monster.

I tried damaging a wall through LineAttack(ACS) and, while it does reduce the wall's HP, it won't activate its special.

Barrel explosions activated with OPEN scripts also don't activate a wall's Death Special, but if the barrel is destroyed with an ENTER script then it will work.

I haven't tested sector damage or Damage Specials.

The WAD contains 4 walls, one with a barrel destroyed by and ENTER script, one destroyed an OPEN script and one with a switch that fires a LineAttack.
The other is just for shooting to show it works normally.