Lute vs. Zither

From what I can tell, then, a cittern or cithern is more like a lute while a citer or cither is more like a zither - two very different things! Observe the cithers laid flat upon the table. Obversely is the cithern held aloft. Therefore it would make no sense for Plato to hate lutes yet love citherns and I was wrong in oh so many ways to "correct" things.