How do I specify in the quest which artifact needs to be obtained?

Set KIND_MANUAL, and write the following in the script:

function artifact_objective()
while 1 do
  if HasArtefact("Script name of the hero") then
    SetObjectiveState("Script name of the quest", OBJECTIVE_COMPLETED);
    break
  end;
sleep(2)
end;
end;

startThread(artifact_objective)

It's easier to write this in the script than to mess around with setting up quests in the editor.