How does the iPhone Upgrade Program work?
Option 2 in your question is the correct one. In the "Getting the latest iPhone has never been easier" section of the upgrade program site you posted it says you have to pay 11 monthly payments (you might as well think of it as 12 months because the upfront payment is in essence your first month). So when you have paid the monthly installment 11 times, you have 2 options:


Turn in the phone for an upgrade while restarting both timers (the 11 monthly payments + 1st down payment and the 20 month full contract).
Continue paying for the phone until the 20 month contract is done and the phone is 100% yours.


The important thing here is that you are never paying for your old phone once you upgrade it.