This is not complicated - though some get the false impression that it is. The key here is "PROOF" that will stand up in a court of law should that become necessary. When it comes to repayment of an over payment of wages, I've never heard of it ending up in tax court - much less seen it.
If you were "IN FACT" over paid in 2019 and did not pay the money back in 2019, then the W-2 is correct. It's a fact that you *WERE* paid the amount reported on the W-2. Period. End of Story. Further discussion on that is an absolute waste of your time. You *WERE* paid the amount in 2019, reported on the 2019 W-2. You will report the W-2 exactly as printed and you *WILL* pay taxes on the entire amount. There is "NOTHING" that "ANY" lawyer can do to change that fact, unless they know something about time travel that you and I don't.
If you pay back the over payment amount in 2020, the payer your reimburse needs to provide you *PROOF* that you paid it back, and they need to provide that proof *IN WRITING*. Then it will be no problem for you to report the repayment on your 2020 tax return under "Right of Claim" to reduce your taxable income in 2020. The TurboTax program can handle this Right of Claim just fine under the "Other Tax Situations" tab.
I would suggest that you pay it back with a personal check, so that if proof is not provided, then your cancelled check is all the proof you need. The check needs to be made payable to the employer using the same thing on the "Pay TO" line, as is identified on the W-2 as they payer. That removes any and all doubt once the check clears.
