Emission rights is a permit from the government to emit a certain amount of whatever. Carbon emission rights can be used as a carbon credit, by buying surplus emission rights from a company or nation that has more permits than they need in an emission trading market. American electricity utilities were trying to game this system when Congress considered emissions trading in the U.S. Many of them bought up old coal burning plants that would soon be obsolete and was lobbying congress to distribute the initial carbon emission rights as a percentage of emission from the current carbon emissions.

A carbon credit is something the company or person can buy to offset carbon emissions, usually offered by a private third party. This may or may not be used in connection with emission rights. The government of Norway has e.g. invested in large tree-planting projects in the tropic to offset national emissions and thus become "carbon neutral".

As you can see, your original instinct is right. One is a government permit, the other is a way of doing accounting with emission rights to get to an imaginary balance of zero.