Raising the price of a thing tends to reduce the quantity demanded.

There are perverse cases. Veblen and Giffen goods (higher prices increase consumption). But generally, the relationship holds.

A cap-and-trade or carbon tax system, to work, has to establish how much carbon can be emitted (I'm ignoring for the moment the argument that the answer may well be "none").

A tax or trade mechanism both invoke market mechanisms to allow individual carbon emitters or absorbers to determine how much activity they'll undertake, including seeking alternatives.

If a tax or C&T at a given level isn't sufficient, then the solution is to change the basis level. The result is still (in theory) achieved in the market.

You and ThrustVectoring are arguing two components of the same mechanism.

See also the Jevons Paradox: increasing efficiency (without also increasing prices) increases the amount of a good or resource consumed.

If you want less of something, raise its price. E.g., tax it.

I'm not sure what the distinction you're trying to make is. The total amount of pollution is the sum total of every individual's pollution. In order to reduce the total amount of pollution, individuals have to pollute less (on average). Conversely, if individuals pollute less (on average), then total pollution goes down.