Until very recently, if I had pending inbox notifications I could visit the inbox tab on my network profile to get a preview. Doing so did not clear the notification in the topbar or mark anything as read.
I find this behavior very useful. Often my primary interface to SE is the mobile site on my phone. Some site functions are difficult to use on a phone, including chat and anything having to do with moderation. Other things are easier to deal with. The network-profile inbox allows me to cherry-pick, handling the easy things and/or the things that are time-sensitive that I should do even on my phone, without risking losing track of notifications that I decide to defer. Later, when I'm using a desktop computer, I view all the notifications, handling the ones I haven't handled already and implicitly marking everything as read. I do not mind that I spend a moment looking at a notification for something I've already handled; that's better than clearing notifications for things I haven't handled.
The behavior of the network profile changed 18-24 hours ago. Now, visiting the inbox tab clears all notifications, just as if I had clicked on the inbox in the topbar.
I assume this new behavior was an accident. (I found no discussion here on MSE. No, not this from 2012.) I can't think of a reason to prefer the new behavior; per-site inboxes aren't synched with the topbar either, so why should the network profile -- which has even wider impact than a single site -- be so synched? Further, the network profile's inbox doesn't discriminate between new and old notifications or show a counter; no state changes (other than this new behavior) if you visit that tab. It's counter-intuitive to me that it should now change topbar state, and it's disruptive to my ability to moderate.