Separate notifications from saving in background

Pulls screenshot notification management into its own class rather
than having notifications handled in SaveImageInBackgroundTask.
(Most of) the notifications code is no longer relevant for the new
corner flow.

Bug: 137153302
Test: manual -- no user-visible changes; tested that flow remains
the same in both old and new versions

Change-Id: I9cdf900f867776d97a339ea65f21e893efc4645e