I think you are correct in your presumption that the rear impact protection system has been triggered. I had a look through all the technical literature that I have, and I could not find any instructions about 'resetting' the headrest if it has moved forward. This suggests to me that there is no special process to follow, you probably just need to grab it and push it backwards.
Below is an illustration and description of how the system works, extracted from one of the Phaeton self-study guides. From looking at this, my guess is that if you just sat in the back seat of the car, and grabbed the affected front headrest and gave it a few shakes fore and aft, it would probably return to the normal position.
It might make the process easier (give you more leverage) if you raised the headrest to the highest possible position first. Be aware that the seat controller is smart enough to know that if the seat base is in the highest position, raising the headrest to the highest position will cause it to hit the roof - so, it limits headrest extension if the seat base is in the upper range of its movement. Because of this, you may want to lower the seat base to the lowest position, then raise the headrest to the highest position after that.
Let me know what the result is.
Description of Active Headrest System (Rear Impact Protection)
text and arrows in red have been added by Michael. Note that of the four illustrations at the bottom of this page, only the lower right illustration shows the headrest in the deployed (post-accident) position. The other three show the headrest in the normal position.