The reason why Morgana pool doesn't proc DoT effects is because "DoT effects" doesn't actually refer to DoT effects; it refers to damage dealt by debuffs, which are almost always DoT effects, but not all DoT is done through debuffs.
In Morg pool's case, the ability basically puts down a thinker that deals AoE damage every second. Same case with Swain Q, except it doesn't deal AoE damage, only damage to a locked-onto target. Cass W, on the other hand, is a thinker that places a debuff to enemies in its AoE that deals DoT damage - notice how when people step out of it, they still take damage from it, as the damage is actually dealt through the poison they get when they step into it.
It's like how Rammus R and Sunfire are different; both deal damage to enemies around the user periodically, but Rammus R just flat-out does AoE damage around him while Sunfire places a debuff on nearby enemies which deals period damage. This is noticeable in that multiple Rammsues ulting will stack their damage, whereas multiple Sunfires won't.