Fundamental: Socialism is not a process of becoming socialism. Socialism is the result of a process establishing socialism. You are not yet in a state of socialism while you are in the process of establishing socialism. Socialism is the public ownership of the means of production, not the workers' ownership absent public ownership of the means of production. Socialism is the real-democratic ownership and operation of the entire state/government. The whole People of the entire state/government own everything together equally. That's real socialism. You aren't in High-Heaven in part or short of perfection.
When two people own their two-person business 50/50 and compensate themselves equally, it is not a socialist entity, per se. It is a capitalist entity.
Socialist-leaning or bringing forth socialism is often confused with socialism proper.
"Peace Train: We haven't seen modern socialism: State collectivism, in which the government controls the means of production, is often mistaken for socialism"
"... but limited inequality based upon differential contributions to the economy exists initially." Wrong. Delay is not socialism.
"The levels at which personal needs are guaranteed increase as the socialist economic system matures." Wrong. Again, delay is not socialism.
"... compensation according to effort as determined by fellow workers ...." Wrong. Different compensation does not exist in socialism. Compensation itself does not exist in socialism.
"... creating socialist institutions based upon cooperation, equality and rational planning within capitalist society ...." Wrong. Socialism creates nothing within a capitalist society. Socialism is the state of zero capitalism.
"... political organizing — establishing an effective political party committed to socialism that contests for power within the capitalist political system." The peaceful establishment is the right approach. It does not have to occur within the capitalist political-system.
That said, we don't stop conversing due to misunderstandings over semantics when we ultimately want the same end.