1. I don't want others who have access to my client machine to be able to see a history of what I did online.
2. I don't want servers to be able to know anything about me except maybe my IP address.
It feels like tying these two together under one setting makes them both fragile. E.g. for scenario 2, I don't care whether a web page can use local storage as long as they don't have access to the data between sessions.
I'd much rather have two options - hide from the server and hide from your boss (or whoever). And maybe some UI to help me always hide from specific servers or delete all the artifacts from a specific session after the fact.