Well, according to the article it is not relevant to distinquish between User Experience and Customer Experience. Instead, Kim argues there are three relevant levels of experience 1) interaction level, 2) journey level and 3) relation level. It’s important that the customer experience is positive on all three levels.
This visual representation can help you clarify who is responsible for what piece of the pie in your organisation. Often both CX experts, as well as UX experts, create journey maps for example. How do they relate?