Fuochista Carbonaio

If a nontoken creature enters the battlefield as a copy of this creature, the copy's enters-the-battlefield ability will still trigger as long as you cast the spell that became that nontoken creature from your hand.

If a token is created that's a copy of Coal Stoker, the token wasn't cast, and Coal Stoker's ability won't trigger. Similarly, if an effect copies Coal Stoker while it's on the stack, the copy becomes a token, but the copy was created on the stack and not cast from your hand, so the ability won't trigger in that case either.

Coal Stoker's ability isn't a mana ability. It uses the stack and can be responded to.