I think you fell for the typical right wing propaganda trope where socialism=marxism=communism. You presented the textbook definition of that propaganda and of communism.
Except not equality - stepping stones. Socialism is supposed to lead to communism, per Marx. That’s not a right wing take, unless you consider Karl Marx a conservative.
Not all socialism is communism, but all communism is socialism.
Notably, the definition given by Marx and other 19th century socialists (worker controlled production) is doesn't even have to involve government or the state at all. It just came to mean that once proximity to power came into play. The "soviet" word merely meant local worker councils, not a state form, for example. But once the Bolsheviks took state power, the power of the actual soviets was successively demolished in favour of a centralized one party state.
So, notably: Self-called "Marxist-Leninists" in the USSR, and Social Democrats (or "democratic socialists" in the US), and Right-wingers all far prefer the term to mean "government controls things." The latter because they can then point fingers and call things authoritarian (but somehow exempt corporate power from critique).
But it is not at all clear that 19th century socialists .. from Proudhon through Marx etc -- who coined the term -- meant this at all.
Also this hard line between "communism" and "socialism" is fictitious and very schematic. All communists call themselves socialists, and almost none of them agree on what that means.
Social democrats trying desperately to sound respectable clearly want a nice hard line there. But they also have no intention of bringing about socialism.
You are a rare person.
Leninism ≠ Stalinism ≠ Maoism ≠ Socialism ≠ Communism.
These are overloaded terms at this point which is part of the reason so few actually comprehend.
Even Capitalism, Communism, and Socialism can mean different things depending on the context and interpretation which is another part of the collective confusion.
If we had clear definitions, it *might* irradicate the confusion, but I don't see that happening in the forseeable future.
There's too much at stake for every involved party to clear things up.
this is perhaps the smallest of all possible reasons that "few actually comprehend" :) yes, the left is good at eating itself, but that self-destruction is not without a lot of external assistance!