Content deletednotification is not triggered when deleting content from backoffice. If the content is deleted programatically by _contentService.Delete(node), then the notification is triggered. Content has variants and the application uses Umbraco version 12.2. What could be the issue?

The deleted notification is only triggered when you empty the recycle bin, you should probably hook into the Trashing notification.

I wanted to remove the content from a third party index when a content is deleted. Whic content notification should be used

Deleted.. or Trashing.. depending on what you think is correct. When you use Trashing, the content can be restored from the trash so you'd need to handle that as well, if you use Deleted then you need to empty the recycle bin for the event to trigger.

Which one is trashing notification. I dont see such notification in the documentation https://docs.umbraco.com/umbraco-cms/reference/notifications/contentservice-notifications

ah you're right, it was renamed to

ContentMovedToRecycleBinNotification

Thanks. It worked.