For Lender X, you are correct that you would want to use the original mortgage date of 1/31/2020 for when you bought the home. For the final balance, that would be the amount of principal that was still owed when you refinanced on June 13, 2021. If you cannot find this on any closing documents of the refinance or your account with Lender X, you can use the same Box 2 Outstanding Mortgage Principal amount from Lender X. The actual amount would actually be a little lower than this for the payments you made from January through June, but it should not impact the calculation if your mortgage is under $750,000 and only slightly if it is above that.
When TurboTax asks when did you buy the home from lender Y, you will want to answer with the date of the refinance, 6/13/2021. As you said, the loan balance for Lender Y should be easy to get from a recent statement that would have shown the balance on January 1, 2022.
