EDITED 3/18/23
Apparently you did it correctly.
See page 27 at https://www.irs.gov/pub/irs-pdf/p970.pdf and page 32 at https://www.irs.gov/pub/irs-pdf/i1040gi.pdf
Filing an amended return, isn't necessarily "wrong". But, you can avoid interest and openly if you can get the IRS to accept your 2022 return, as submitted. I would try calling the IRS at the number on the letter.
Q. The two tuition reimbursements from my employer are both tax free because I received each of them in separate tax years and each reimbursement is under the $5250 threshold, correct?
A. Correct.
Q. . Because I received the 2022 reimbursement that was for 2021 tuition, I must re-calculate the lifetime learning credit (LLC) and pay education recapture tax, right?
A. Half right. You must re-calculate the lifetime learning credit. But, this is not a recapture situation. You do not report anything on your 2022 tax return.
Q. If the answer to B is yes, what should I do with a $906 I received from the IRS after I filed my 2022 taxes? Should I keep it or should I use the online IRS system to pay it back?
A. Keep it. The IRS was correct to refund it to you, as your 2022 return was technically erroneous. You must prepare an amended 2021 tax return, eliminating the LLC, and send them a new check.
How to amend https://ttlc.intuit.com/questions/1894381-how-to-amend-change-or-correct-a-return-you-already-filed
I'm curious. How did you report the $906 on your 2022 return? What form, what line?
