Status confusion cycling between Deactivated and Activated

The sqlite-database-integration plugin, standalone, doesn't show activation and deactivation status correctly.

To reproduce:

1. Activate the plugin.
2. Look at the Plugins -> drop-ins page (or just the drop-ins count) to ensure wp-content/dp.php is in place.
3. Look at whether the plugin shows as Activated. Sometimes it wrongly shows as Deactivated.
4. Activate it again.
5. Look at whether the plugin shows as Activated. It should show that way.
6. Deactivate the plugin.
7. Look at the Plugins -> drop-ins page (or just the drop-ins count) to ensure wp-content/dp.php is gone.
8. Look at whether the plugin shows as Deactivated. Sometimes it wrongly shows as Activated.
9. Deactivate it again.
10. Look at whether the plugin shows as Deactivated. It should show that way.

The root cause of this: most likely the fact that the active_plugins option gets updated in the database that's in play when starting the activation or deactivation workflow, but not in the database that's in play after the workflow.

(This isn't very important in production, but it's confusing when testing things.)

Trace the plugin activation and deactivation workflow, focusing on where the active_plugins option is read and updated when the active database changes. Reproduce the sequence using the Plugins → drop-ins page and check wp-content/dp.php and the drop-in count after each transition. Done means repeated activation and deactivation consistently show the correct status.