The first scenario you listed is correct.
You choose a target and make an attack roll. If it hits, you roll for damage. Then you choose another target (the same target or a different one) and make another attack roll. If it hits, you roll for damage.
The 'Extra Attack' Feature simply allows you to do two attacks (or more as a Fighter levels up) whenever you spend your action on a turn to attack. This means that you make two completely separate attacks, not a single attack that you can roll twice on (as the third scenario suggested). Additionally, the second attack is not influenced by the result of the first attack, as the second scenario suggested. Regardless of whether you hit or miss the first attack, you can do the second attack.