Socialism means the state has social support networks, but largely works under capitalist rules

What you're describing is "social democracy" — capitalism with safety nets, where production is still controlled by owners rather than workers. "Socialism" explicitly implies worker control of production. "Nordic socialism" could more accurately be called "Nordic social democracy."

"Communism" refers to a classless, stateless society where everyone has what they need, no one is exploited or coerced, and there are no wars. It's an aspirational vision for the future, not something you can do right after a revolution when capitalism still rules the world.