In the way Americans use these words, that honestly doesn't sound that far off. But that's a difference in definition similar to how liberal means different things in the US vs everywhere else.

In a European context I would have said that capitalism is companies being owned by private owners, communism is companies being owned by the state (which is supposed to be the extension of the collective will of all citizens, but rarely is). Socialism is whatever you want it to be, but generally includes the state supporting the unlucky, improving worker mobility through unemployment benefits, improving worker rights, supporting the young and the elderly, etc.

The US and EU have some differences in how they approach infrastructure, communication and transportation, but in nearly all other aspects they both bet fully on capitalism. The difference is that the US is very big on free unregulated markets within the US (though with some comically protectionist policies when there's the threat of competition from outside), while the EU is generally of the opinion that a regulated market is a better, more competitive environment with better outcomes.