You said that you had a leased vehicle that you traded in for money of the MSRP price of a new vehicle that you are now financing.
When your lease is up on your vehicle, you have no positive equity or any value in that vehicle. So, you cannot use the leased vehicle to trade-in towards a down payment for a new vehicle. Some how you and the dealer miscommunicated with each other over your vehicle being financed or leased.
Now that the dealer has taken in your vehicle as a trade, and should have taken it in as a lease return, because they now found out that the vehicle was leased and not financed, they want their money they gave you towards your new vehicle.
If I'm not right, then you need to show us your paperwork so we can figure it put for you.
Understood, but that doesn't explain how OP had equity in the leased vehicle. If OP would have turned the vehicle in early, OP would have negative equity in the vehicle and that payment would be added to his financed vehicle. From OP's statement that the dealer told him to "come and pick his old vehicle up", makes me believe that it is a used car dealership not under a national brand and not a new/used car dealership like Chevrolet ect...
All national brand U.S. dealerships accept lease returned vehicles that they will hold in their yard until the finance company picks the vehicle up for auction (unless that dealer doesn't buy that vehicle themselves). A used car dealer not under national brand, will not take in lease returns. That is what leads me to believe that the OP misrepresented his vehicle as financed and not leased. Once the dealer called the finance company for the title, they were told that the vehicle was leased and needed to be returned. Now the dealer wants their car back that they sold to OP, or the equity from the leased vehicle they gave OP towards the down payment for his new vehicle.
I cannot think of another situation that OP would be in if that vehicle was leased.
Funny, you left that part out so I included it for you.