If your job was located in NY, all of your income is subject to NY tax. Your employer was correct to report all of it as "NY income" in boxes 15-17.
If you moved to CT during 2020, then what you earned in NY after the move is subject to both NY and CT taxes. You will have to calculate the amount earned, possibly with the assistance of your employer's payroll department. You can also ask them to add CT as a reported state on your W-2, but NY will still show as the source of all your NY earnings.
You will have to file part-year resident returns for both states. Complete the NY return first so the program can give you a CT credit for taxes paid to NY while you were a CT resident. This is how the states avoid double taxation by allowing a state credit in your home state.
This post provides additional instructions on how to file in two states You moved to a different state during the tax year?
