It’s amazing how people just make things up. I genuinely have no idea where you got these definitions unless it was some hole on Reddit or similar.
What manages the means of production if not a government? Saying “the people” is as hollow as the US talking about “Freedom” and “Democracy”. “The people” cannot merely project their will into the aether and have it realized, they need some method of organization. They need to be able to administrate complex systems rather than just hang out in “primitive communism but with high technology somehow”. Whatever that system is and whatever you call it, that’s a government. In a system of democratic government that administers things, the difference between “the people” owning things and the government – here an organ that exists only so the people can manage the means of production – owning them is immaterial.