2002 Cadillac Deville Transmission identification

I hit an object in the roadway which destroyed my transmission. Insurance is paying the cost to replace it with a used transmission. Repairs are being done by a Cadillac dealer. The first transmission sent had excessive mileage and was burned up. When opened, they found no filters, burned fluid, etc. It was rejected. I get two stories regarding the second one. 1: The service writer at Cadillac tells me the second transmission was for an Oldsmobile and was refused. Therefore, a third transmission is required. 2: The insurance adjuster states that he sent the correct transmission the second time and the technician installing it, dropped it and cracked the case. Therefore, a third transmission has to be acquired at Cadillac expense.

How can I identify a transmission as being for a Cadillac versus an Oldsmobile and what transmission is the correct one for my car?

I need to identify who is lying to me. I don't know who to trust.