This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the functional calculus and that became more common after the Banach algebra revolution of the 40's.
Circa 1960 Dunford-Schwartz were still mainly resolutions of the identity.
By the time reed and I did Reed-Simon in the early 1970, we preferred to talk about multiplication operators emphasizing the individual spectral measures which was in the air when we wrote. Who first using that language and who made it popular?
Best Answer
I do not know who used it first, but I claim that Halmos made it popular in his 1963 Monthly paper.
There he makes the connection to the diagonalization of hermitian matrices and mentions that people usually called "multiciplicity theroy" this part and not the spectral theorem.
Multiplicity theory is described in great detail in his 1951 Hilbert space book.
Finally, Dieudonne claims in his History of functional analysis book that this connection was maid by von Neumann after the development of the Gelfand-Najmark theory in 1943, but he does not give an explicit reference.
ADDED: After reading the answer of Francois Ziegler and Igor Khavkine, let me mention the following. NEumann in his 1931 Ann. Math. paper had all what you need to formulate the spectral theorem in this form. Stone in his famous book, Theorem 7.10 formulates this statement with proof. Thank you for following this up.