plugin-manager: Don't try to starte effect when shutting down

Add internal state (starting, running, stopping), and use this instead
of MetaDisplay struct fields to determine whether to start animations.

This fixes issues when we try to animate things when shutting down.

Part-of: <https://gitlab.gnome.org/GNOME/mutter/-/merge_requests/2716>

(cherry picked from commit db1000e2)