I would broadly agree with your description of the philosophical kind of conceptual underpinning of carbon pricing, with one caveat, and that is that the carbon pricing applies beyond the private sector. You're right that it will apply to households. As householders, you and I will make decisions based on whether we will invest in energy efficiency, change our windows, choose what kind of car we will buy, or decide to take transit.
Similarly, the public sector is a large consumer of fuel and goods. The public sector will also face incentives provided by carbon pricing—hospitals, government buildings, etc. The idea is to apply the price as broadly as possible across the economy, but again, as you said, then the idea is to provide complete flexibility as to how to respond to that price signal.