The basic idea of carbon pricing is to force people to take into account more of the consequences of their own actions. This is essentially a response to market failures; while there is an incentive to overfish from a commonly-accessible lake, it doesn’t make sense to overfish your own pond. That being said, there do seem to be two possible philosophies for setting the price. One approach is to approximate the total level of harm your actions cause, then set the price at that level. But turning the suffering of others into your suffering, you are given the incentive to minimize the harm, and avoid actions that have more harmful than beneficial consequences. In the context of climate change, however, the overriding task is not simply to emit less, but rather to build a society where people live carbon-neutral lives. Arguably, then, the level of the carbon price should be set so as to make that transition maximally easy, while making sure it happens fast enough to avoid abrupt or catastrophic change.
In the end, this distinction is largely academic, given the uncertainties involved. (There is also the matter of choosing a discount rate, which has enormous effects on your estimate of the net present value of harms that occur in the future.) Since the transition to a carbon neutral society will be a one-off change, we will probably never have the information required to know for certain whether it could have been done more quickly or efficiently. Once the risks associated with inaction are taken into account, it seems clear that we need to establish a carbon price that spurs rapid action and a sustained commitment to economic transformation.