What you’ve just described are socialized safety net programs inside mixed capitalist economies. That’s not socialism. There is no private ownership in socialism. Consequently, there are no rich in that system. The “community” owns and produces everything.
Communism is about the absolute equality of all - no rich, no poor. Socialism is about caring for the poor, with no upper bound on the rich.
In other words one squeezes from the top down. The other pushes from the bottom up.