Yes, you are correct.
Tableau server -no student version
tableau server, we can use the same key on 3 installations ( dev, test, prod).
You're right, the Tableau for Students program includes Tableau Desktop and - since May of this year - Tableau Prep. There is no student version for Tableau Server.
Further details are in the FAQs below...
Also you are correct in saying the Tableau Server license allows up to 3 activations (so one production environment and then two non-production environments which could be Dev or QA etc.)
Hope this helps
[Program Manager - Tableau Community]