But many Socialists don't define the terms in that way. Choosing appropriate definitions of terms is a tricky and sometimes confusing process. I can't say much about it here, except to say that if the question is to be answered, if one really wants to understand how socialists have reconciled these things, it's important to keep their own definitions in mind.
In the United States, people who don't consider themselves "socialists" mostly identify "socialism" with government control of the economy. People who do consider themselves "socialists" mostly don't. They may support government control -- some do and some don't -- but that's not how they define socialism. W. A. Lewis, Nobel Laureate in economics and at one time a spokesman for the Fabian socialists, said that the essence of socialism is democracy and the classless society. Perhaps that would be a better definition of "democratic socialism," but almost all socialists have agreed that socialism has to be democratic, even if some of them have twisted the meaning.
There were critics of government control within socialism from the very first. They made two practical objections to government control, among others: 1) bureaucrats might very well form a new class, living on the labor of others not through ownership but through taxation and salary, and 2) an economic system organized that way wouldn't work very well.*** The experience of the Soviet Union does seem to support those points, which were made by some socialists right from the beginning.
In Lewis' sense, the Cooperative Commonwealth could be a form of socialism. When enterprises are democratically controlled by their employees, there is no function for a class of rich employers. The existence of a class of rich employers conflicts with the ideal of reciprocity -- domination of employees by employers is anti-reciprocal! So a fully developed Cooperative Commonwealth would be a classless society. Also the control of the enterprises and of the associations among the enterprises (that is, the control of the commonwealth) would be democratic. So the Cooperative Commonwealth would fit Lewis' definition of socialism.
Since I define socialism as Lewis did, I would agree that the Cooperative Commonwealth would be a form of socialism -- a highly developed and perfected form of socialism.