If there is confusion, it is most likely the confusion of the ideas of "murder" and "killing a person." To say that
murder = killing a person is oversimplification. While the definition would surely vary between societies, "murder" generally is defined as:
the crime of unlawfully killing a person especially with malice aforethought
(source: Merriam-Webster. Note the use of the words "crime" and "unlawfully", which would have their own definitions that would vary by society.)
In other words, "murder" generally is not entirely synonymous with "killing a person" since it involves specific details, such as being unlawful. If capital punishment is lawful, then it would not make sense for it to also be murder, which is unlawful.
The following argument contains the fallacy of "affirming the consequent":
If there is murder, then a person killed another person.
A person killed another person.
Therefore, there was murder.
Using the above definition, it is entirely possible that there is "killing a person" without there being "murder." We could teach our children that murder is wrong without teaching them that killing another person is always wrong.
In fact, practically speaking, societies generally don't teach their children that killing another person is always wrong. Most societies have some form of military, some type of law protecting killing in self-defense, and sometimes a legal distinction between "killing" and "premeditated killing". So, if we are already teaching our children that some killing is not wrong, such as serving in the military, then what would prevent us from teaching them that capital punishment is another instance in which killing another person is not wrong?
There could be confusion, but it is a kind of confusion that needs to be explained using the proper definition of "murder." However, to say that capital punishment for murder is a paradox is a false statement (using the generally accepted definitions).
A paradox (of sorts) would be if an opponent of capital punishment said, "Capital punishment is wrong, and the people who perform it should be executed for their crime."