As the notification redesign affects the API anyway, we can just
as well use the opportunity to modernize the code.
Turning the icon into a GObject property means we no longer need
a custom signal for change notifications, and the icon becomes
usable in bindings.
Since setting an `GThemedIcon` is common this also adds a convenience
property to set the icon name directly.
Part-of: <https://gitlab.gnome.org/GNOME/gnome-shell/-/merge_requests/3103>
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