Both of you are over-simplifying too, although Constitutional Armory takes the cake with his buffoonish view. The term "own" is not a single thing that turns on and off like a light switch. "Ownership" is a bundle of rights, which can be separated, and come into effect at different times.
If you enter into a contract to buy something (house, car, gun, loaf of bread), you have certain rights. For example if the other side is unable to fulfill the contract, there are laws about how to unroll the contract, and who can get what.
If you pay for the the good, that gives you additional rights. Undoing the contract and the payment gets more interesting now. Interest may start accruing.
And for goods that require registration or title (house, car, gun, ship, airplane): until that paperwork is done, some of the ownership rights don't actually transfer, but others do.
One of the examples I remember from civil law class is this: A shipping magnate M buys a used oil tanker from shipping company C, tonight at dinner while having after-dinner drinks. They agree that it will cost two million, 50% down payment. Inspection and transfer of the ownership to occur next time it is in port, but no later than July 1st, and seller to meet buyer at the Nassau (Bahamas) "DMV for ships" within 48 hours afterwards to file the papers. Second half of payment due at the DMV office, with a bond posted in the meantime. They shake hands, and have another drink. Tomorrow, M wire transfers the first million to C. Now let's look at a few cases: (a) the ship had already sunk last night, (b) the ship sinks next Wednesday, after the contract was entered and the down payment sent, (c) the ship sinks right after inspection/transfer on the way out of the harbor, (d) the ship is OK, but the big engine just seized and it is stuck in Rotterdam for the duration. Turns out that at no point from the meeting of the two gentlemen to the DMV appointment either of them fully "owns" it, but the various parts of ownership have to be separated into bits and pieces, and all unrolled according to the rules. Discussing this case takes a whole 2-hour lecture, since this stuff is complicated.
So in these edge cases like what we're discussing in this thread, you can't just say "I own" because "I paid for", it's more complicated. To figure out the detail, you have to consult a lawyer (which is NOT me). On the other hand, civil law and ownership transfer is one of the most fundamental parts of the law, so anyone who practices law will know the answer in simple cases.