Your definition of socialism is more akin to a definition of social democracy, which is... maybe a form of socialism, depending on who you ask -- it is historically contentious and generally accepted that social democrats aren't socialists.

Socialism can have all of the things that you described, but it is decidedly anti-capitalist. It reorients how workers relate to the means of production. Under capitalism, the means of production are owned by the bourgeois class, while under socialism, they are collectively owned by the workers.