In "Operator algebras with a faithful weakly-closed representation" (1955), Kadison describes a countable W*-algebra as a C*-algebra which has a faithful representation as a countably decomposable ring of operators. Unfortunately I was not able to find any other occurence of this structure in an other article or book. Have you already seen this notion of "countable W*-algebra" somewhere else?
If I'm reading it correctly, "countably decomposable" means that there is no uncountable family of mutually orthogonal nonzero projections. Such von Neumann algebras are now called "$\sigma$-finite".