I think what you're trying to get at is unlike most wars, whether they are declared wars or "police actions", there is usually a treaty that officially ends the armed conflict. As the CSA government disintegrated without signing or declaring any suchvend, and its armed forces surrendered piecemeal, there is no traditional end.
Since the USA refused to recognize the CSA it raises the question of, if Jeff Davis and the Confederate peace commissioners had accepted reality and agree to end the war on terms Lincoln would accept how would the surrender document have been worded? "As representatives of the states engaged in rebellion"?