I think it is easy for us to agree in that, from the client's point of view, logging out of a website is idempotent.

Now, you got a point about idempotence from the server's point of view. However, it would take a _badly_ programmed website for the logout operation to _not_ be idempotent. Sending notifications, updating counter, etc. _without first checking if the user is really logged in or not_ is simply moronic. This simple check is what would turn the logout operation into an idempotent one in the server too.