your employer is correct. if you had paid it back in 2020 you would have gotten a corrected w-2. since the payback was in a subsequent year IRC section 1341 is what controls your reporting. you do not amend 2020. When you do your 2023 return in 2024 this is the procedure:
You can either take the repayment as a deduction on Schedule A line 16 and label it as "IRC 1341 repayment" OR
recompute the taxes for 2020 by excluding the payment. The decrease in tax is taken as a credit on the 2023 Schedule 3 line 12d and labeled as "IRC 1341".
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note since the forms for 2023 are not out yet the location to report may be wrong.
