How should I thank this person who wrote a letter of rec for me (but to a different school than the one I'm attending?)
Directly, as if they were an actual human being.

"Thank you for providing letters of recommendation for my graduate school applications.  I will be joining the musicology PhD program at the University of Southern North Dakota at Hoople next fall.  Although I did not need your letter for my USNDH application, I still very much appreciate your support through the application process."