There is an OEIS entry for the triangle of isomorphism classes of size n for each order m of magmata: A079171. Notice the second entry of the fourth row, which should correspond to those magmata of order four with precisely two distinct labelings: the entry is a 1, which means there is a unique such unlabeled magma.
What, pray tell, is this magma? I have no idea how to approach this question.