The cluster's name and url should match the default settings of the cloud provider for the provider detection to work correctly. Simply do not mess with your local config file and you will have no issues.
Usage
Click the K9s icon in the sidebar.
Click "Open K9s" in the panel.
K9s will open inside a new terminal editor tab.
Left-click on a kubernetes cluster to switch the active context to it. right-click to copy it's name.