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.
Capitalism is ownership of capital by corporations who are protected by the state.
Neo-liberalism is a version of capitalism that thinks the state sucks and must stick with the minimal protection that enables private ownership (vs banditry).
Communism is ownership of capital by the state and the abolishment of private ownership.
Socialism is more fluid, but the core is redistribution of wealth to effect just outcomes.
All of the above involve the government applying rules so 'something' doesn't get out of control, whether its evil bureaucrats or evil corporations. All of these can be done democratically (usually in varying degrees) and with the rule of law, or end up in a form of oligarchy or lawlessness.
Capitalism is government regulation of a market. https://youtu.be/CRPHp2EjNR8
Capitalism is like a cancer, it will grow endlessly and will become feudalism. Socialism is the government applying rules to capitalism, so it doesn't get out of control.
US policy of neo-liberalism is growing in Europe too, which is giving problems.