I guess to be fair, Kagi does have two different AI components. They have their Kagi Assistant which is your bog standard AI chat interface, which does cost extra and is displayed in the details page you are talking about.
But they also have Quick Answer which is like the google AI overview that appears when you make a search. It will summarize the results for you. It auto triggers when you end your search with a ? or if you click the Quick Answer button. It can be disabled. However it does not get charged extra for, it is part of your subscription.
So its kind of both.
From what I'm reading, your account and followers are attached to your devices, meaning you can easily point at a different relay and be back to normal. It's a neat design honestly