RE: Axiom-1 mission, It is the first private mission.
I think an important part of the story to take into consideration is that this is a commercial mission provided to a public space Agency. The plan for commercial companies providing services to NASA still involves contracts raised by the agency to be provided as a service by companies, in opposition to the provision of materials and assets (spacecraft, launchers, ground segments...). In the same way the future commercial space stations will still provide a service to NASA, so in the end it is a small change on the final property of the assets, but the payer will still the same.
However, it brings some good news, more companies will bring new projects and capabilities, eventually.