Marxs definition of socialism is the important definition because it’s his word. Marx did think socialism would be the next step after capitalism and that communism would eventually follow. But he thought communism would follow in a far off future.
“The United States regulates businesses, provides welfare. Those are socialist ideas. China, controlled by the Chinese Communist Party, has rich and poor people. It isn’t communist.”
Socialism is not when the government does stuff. I know you have been conditioned to think that but that’s not what it means.