I agree that the bell icon is a commonly used method of altering users to notifications. I'm not sure it that would be classed as a push notification though ...
Correct, it is not the push notification - that's something separate.
The push notification appears as a transitory pop-up. Now that I've finally enabled it, it shows in the lower right corner of my primary desktop window, above the time and date in the tool bar (the same place new mail notifications from Thunderbird appear). The pop-up appears regardless of whether my browser has focus at the time. (Firefox under Win10). Your mileage may vary with different browsers or operating systems. Maybe also depending on where you have the tool bar docked.
The bell list, or notifications merged into messages list, is a UI feature for finding current and prior notifications.
In a typical "bell" list there's a visual cue for which events you've "seen" and which you haven't. That's functionality that doesn't currently exist in the Messages list, except by the incidental link coloring done for visited pages (which uses browser history to distinguish seen from unseen). But that functionality is a long-standing request that Mark has often mentioned in conjunction with the notification overhaul.
Aren't these initial changes part of an overhaul that paves the way for release of the long-awaited mobile apps?
That's my understanding: this is the web version of functionality to be added to the mobile app betas. Presumably in the mobile app the push notifications would go to the device's notification display mechanism, whatever that is.