Part of ENG-4844
Depends on D9108
Enabling web database for all users, including non-staff and anonymous users. Merged clearSensitiveData with init because they need to be done atomically: clearSensitiveData leaves db in uninitialized state, which caused problems in some edge cases. Also renamed notSupported to notRunning as it better describes the db state. Also simplified web/sqlite-data-handler so it has structure more similar to the native/sqlite-data-handler.
Draft migration handled in the next diff.