They were following the orders of their immediate commanders, just like any soldiers are supposed to do. Prigozhin's "chain of command" at that moment was part of the government's chain of command
Meaning they were answerable to Prigozhin, not Putin during the revolt. But even more fundamentally (apart from what happened during "revolt time"): as an organization and in terms of the laws that applied to it, it was entirely separate from the regular Russian army (and supposedly didn't technically exist on paper at all). That's why it's official name was PMC Wagner, where PMC stands for, drum roll... Private Military Company.
I'm really not sure what you're trying to get at with this "not a private army" angle, when it by all indications it was exactly that, or with the idea that Prigozhin was just like any other commander in a regular army, when of course he wasn't.
But that's okay, you can model and label this information however you want.
>why it's official name was PMC Wagner, where PMC stands for, drum roll... Private Military Company.
Yes, and North Korea is a Democratic Republic.
>by all indications it was exactly that
There is no such indications, only name. Other than name - PMC Wagner had nothing to do with a private army. It was a government-organized, government-funded, government-chai-of-command-integrated experimental military force
>idea that Prigozhin was just like any other commander
Rather something like government-appointed director of the special service
It was a government-organized, government-funded, government-chai-of-command-integrated experimental military force
Whose members shoot down government helicopters and an airborne command center when its government-appointed director orders them to. Following the government-chain-of-command, we imagine.