The Published Trigger fires when the associated Asset or Content is pushed to the queue. For example, if you have a Published Trigger listening to an unlisted YouTube video, when Conducttr publishes that video and changes its privacy to Public, the Published Trigger will go to True. BUT if you manually go into YouTube and change the privacy yourself (i.e. not via Conducttr) the Published Trigger will not fire.
Possibly the best example use is for situations when the audience unsubscribes from a character. If an audience member sends an SMS with the word ENOUGH as the match phrase and this SMS trigger is connected to an Unsubscribe trigger, an SMS reply will never be sent because due to the trigger processing order (see Figure 78) the audience will be unsubscribed before a reply SMS can be sent. To avoid this, use the SMS trigger to publish a reply and now use a Published Trigger attached to the reply to trigger the Unsubscribe action.
Another use is for when message content is published after a (long) delay if that delay period signals the beginning of something new. Using a Published Trigger attached to the delayed message provides a convenient way to trigger the next sequence of events.