Coppelia needs to be sent to the Graveyard by an opponent's card. It's not being sent to the Graveyard by Share the Pain. It's being sent by a game mechanic, similar to how a card like Rivalry of Warlords works.
Terror Incarnate isn't as picky about how it leaves the field. As long as it doesn't miss its timing (and doesn't get banished face-down/returned to the Deck), it'll trigger.