Also the king is special like the CL (who also kills).
When you’re killed or executed, the result is the same, rendering the difference entirely cosmetic, so King is a role that can choose to kill - and is a ‘killing role’ by the standard you set forward that any role that CAN kill is a killing role. We done.
An execution is different because it is public and it requires others to Agree. Even the Kings decide fate requires enough people to vote the person up in the first place to work. And even then the Decide fate is both 1 time use and also can be used defensively.