It specifies movement on your turn, or at least heavily implies it.
You are absolutely right. "On your turn, if you move at least three squares from where you started your turn..."
So it would trigger if you take down a cursed enemy, but not if a cursed enemy goes down when it is not your turn.
My point was that any mode of movement will trigger Shadow Walk including teleportation (such as... feylock pact boon and feystep), jumping (with athletics), climbing (with spider skitter?) etc... its all good as long as it is on your turn.