In his book on commutative rings (published 1970), Kaplansky talks about Macaulay rings. In the mid 1970's, I learned some commutative algebra from a student of his, who referred to these rings as Macaulay rings. When I returned to commutative algebra about twenty years later, these rings were now Cohen-Macaulay. When did the change occur?
Actually, the name Cohen-Macaulay rings and Macaulay rings areboth due to Zariski and Samuel (Commutative Algebra, Volume 2, App. 6, p. 396, 1958):
Definition 1. Let $A$ be a local ring. The common number of elements of the maximal prime sequence in $A$ is called the hological dimension (or the grade) of $A$, and is denoted by $codh(A)$. If $codh(A)=dim(A)$, we say that $A$ is a Macaulay ring (or a Cohen-Macaulay ring). [emphasis mine]
So, it was Kaplansky's decision to call them just Macaulay ring (see, e.g., Irving Kaplansky, Commutative Rings, Chapter 3, p. 84, 1974), but at the same time other authors used the name Cohen-Macaulay ring, e.g., Mario Fiorentini, Esempi di anelli di Cohen-Macaulay che non sono di Gorenstein (Examples of Cohen-Macaulay rings that are not Gorenstein rings), Atti della Accademia Nazionale dei Lincei, Serie 8, Vol. 50, n.2, p. 94–99, 1971. (in italian)