Is the Escrow refund check considered proceeds of a home sale to be reported in box 2 of Form 1099-S

No, the escrow refund check is not part of the proceeds from the sale of the house.

The escrow refund check is the money remaining in the escrow account after the payment of property taxes and/or insurance. This is what you paid in excess into escrow. This refund is a refund of your own money and is not reported on your tax return.