Jumping in to hopefully clarify something. The anarchist definition of the state is different than the Marxist definition of the state.
The anarchist definition of socialism is also different than the Marxist definition of socialism. Generally speaking, to anarchists, socialism and communism are synonyms, and there really isn't the lower/higher phase distinction.
State capitalism is a term used to describe the economic systems of places like the USSR. The state steps in and becomes the capitalist, in essence. The worker is in a similar position of not really owning the means of production, in the same way that the public doesn't really national parks in the US. In paper, in theory, and perhaps in spirit, the workers in a socialist state own the means of production, but in reality it is owned by the [the party/the state/an elite group of people]. There is still a similar incentive towards growth, there is still a group of people profiting off the backs of those who do the actual work of creating the items needed for survival, and there still a disconnected between those who do the labor of keeping all of us fed and clothed and the use of those things. Workers are not directly in control, and that's the problem, to the anarchist view.
Effectively, the anarchist is view that we can and should move directly from our current system to a stateless (by the anarchist definition of the state), classless, moneyless system, without an intermediary state in between.