Socialism — the dictatorship of the working-class — is the transitional mode of production between capitalism — the dictatorship of the capitalist class — and the stateless, classless mode of production that is communism. You can't really separate the two.
Thats the communist definition of socialism. Socialism originally just referred to the leftist movement as a whole (including anarchists and dem-socs, which i guess he refers to). And is also used to refer to the concept of workers owning the means of production
The first line in Wikipedia "Socialism is an economic and political philosophy encompassing diverse economic and social systems[1] characterised by social ownership of the means of production".
To be clear, this definition goes back to 1832, where the original inventor of the word used to define a society "based on the shared ownership of resources". So it is not just "the communist definition" it really is the definition, it did not "originally" refer to anything else.