I'm taking a break from my most-used SE site for a while, so I've logged myself out, but there are other sites in the network where I'm still interested in reading and possibly contributing. I'm (obviously) logged in to these so that I can interact meaningfully; therefore I get notifications in the hyperglobal unified supercollider inboxadongdong.* The thing is, my accounts are all linked, and I'm getting notifications for the site which I'm logged out of.
I'm not really interested in those right now. To be frank, I don't want to see them at all. That's why I logged out. Please stop telling me that I just earned a "Nice Answer" badge on a site that I'm trying to avoid. Jokes about Jeff's digital crack are one thing; constantly trying to lure me back is just aggravating.
The inbox should at least not be badged for sites to which I'm not logged in. Preferably the content ("something good happened" notifications and comment responses) should not appear there at all.
Using SE on different computers with different sets of logged-in accounts on each is certainly a point in favor of the current behavior, but it would still be possible to get notifications were this to be changed. If it's not changed, there's no way to disable notifications short of account deletion.
*Apologies, I forget the exact offical funny name for it.