The Hey Cortana app for Windows 10 Mobile has a new update for the Lumia 930 and Lumia 1520 (but not the Lumia 950 or Lumia 950 XL). The update includes some user interface changes and, more importantly, a new training algorithm that should make learning how to use Hey Cortana better.
The new system lets users simply say "Hey Cortana" five times with no example scenarios anymore.
The app also apparently fixes bugs for some users who previously were unable to the training system to work for them. We previously posted a workaround for this issue but it now appears this update makes it unnecessary.
Interestingly, Hey Cortana on the new Lumia 950 and Lumia 950 XL is found under Cortana's Notebook and not in system Settings. Indeed, one other feature with today's update is the ability to choose whether to have Hey Cortana work all the time or only when plugged in and/or the display is on. That technically gives the Lumia 1520 and Lumia 930 one more feature than the Lumia 950 and Lumia 950 XL.
Still, with the newer Lumias you can bark a command at Cortana without pausing e.g. "HeyCortanaRemindMeToGetMilk", instead of saying "Hey Cortana (pause) --command--". Some of these differences are likely due to the different chipsets used in the phones described here.
To grab the update just head to the Store and use the check for updates function. Once downloaded, you will need to restart the phone to get the new features enabled.
Thanks to everyone for the tips!