I think the differences are a lot bigger than what you’re suggesting here. A socialist system has the means of production controlled by the government. A social democracy still has (most*) of the means of production run via a market economy, and just harnesses some of the surplus that it produces to provide things like health care, education, a decent standard of living, and a social safety net.
*Things that are natural geographic monopolies (roads, utilities, etc. ) are pretty clearly better off being run as municipal utilities. Basic research is also something that is apparently done better by a government. OTOH, most other things are not nearly as amenable to centralized control and planning.
Which I am fully in support of, and which project I think is being damaged by calling FDR-like social democrat policies “socialist.”
