Flame Totem with Three Dragons do not proc HoT on kill effect

Herald of Thunder does not trigger on totem kills - this is intentional. You have the buff on you, not the totem, and only the thing with the herald buff on it can trigger the effect by killing shocked enemies.
You can very occasionally trigger it if a totem ignites an enemy and the ignite kills them. Since the totem might not exist any more when the ignite expires, that kill is assigned directly to you.

Totems never are your spells. They are separate entities. They cast your spells. This is an important distinction.

When you use herald of thunder, it puts a buff on you, and not on the totem, that adds lightning damage to your spells. Since the totem is casting your spell, that spell has the lightning damage.
When the totem kills a thing, your totem killed a thing, not you. Your totem does not have a herald on thunder buff on it, and thus cannot trigger herald of thunder's on-kill effect.

You appear somewhat confused on how auras work with totems.