I don't think socialism really exists - it's a point that is unstable and is more or less on the path to full control (communism or fascism.) Also, even in the most leftist countries, say with communism, there are still some markets - they are just controlled and constrained.
It's also confusing because economic explanations don't capture what leftism is about. It's more than "means of production." The scale is levels of control with the fascism and communism to the left, differing by levels of nihilism. The communists are bent on destruction of civilisation and that requires power. The fascists are about profiting from power but they don't destroy from within like the communists (woke leftists) do. For example, the fascists wouldn't redefine genders.
Communism, Marxism, Socialism, Fascism, Progressivism, etc. are all left wing. The right is about limited government, freedom, liberty, free-er markets, etc. - all based on common morals that are taught. (The common morals are important.)
It's even more confusing because it's been the left who has defined the terms for many scores of years - for example, pushing that fascism is on the right. Also in Europe, most systems have been shades of left. In many cases, even the "right" is to the left.