That's correct, but I'm not sure what you understand those terms to mean, because neither really supports taking all ownership away from people. I'm just gonna leave this blorb here, because I feel like this is where it fits best.

Communism in the style of Marx and Engels means that the workers own the means of production. They would have been completely in favor of a person owning their own farm (or jointly owning it if multiple people worked it). They didn't really envision much of a state to interfere, much less own property.

That the Soviet Union (and later the PRC, fuck them btw) claimed to be building the worker's paradise under communism was mostly propaganda after Lenin died. There hasn't been any state that has implemented actual communism as established by theory.

Socialism (as I understand it, but I'm not well-read on it) means the state has social support networks, but largely works under capitalist rules, with bans of exploitative practices. There are some countries trying to implement a light version of this across Europe, to varying success (mostly failing where capitalism is left unchecked).

The issue is that the US started propagandizing like mad during the cold war, and "communism" was just catchier to say than "supportive of a country that is really just a state-owned monopoly". Soon everything that was critical of capitalism also became "communism", which eventually turned into a label for everything McCarthy labelled "un-american". This is also the time they started equating the terms communism and socialism. A significant portion of the US population hasn't moved past that yet, because it fits well into the propaganda of the US being the best country in the world, the American Dream, all that bs. The boogeyman of "the state will take away the stuff you own" turned out pretty effective in a very materialistic society. Although I'm very glad to see more and more USAians get properly educated on the matter and standing up for their rights rather than letting themselves be exploited.

Dude walked right into that one, didn't they?

Communism is when no toothbrush

You individually are allowed to own the means of your production if you don't exploit other people. Everyone who works at the farm owns the farm.