A wider quotation makes the intent quite clear:

When a creature makes an attack roll against you, you can use your reaction to impose disadvantage on that roll. If the attack misses you, your next attack roll against the creature has advantage if you make it before the end of your next turn.

Once you use this feature, you can’t use it again until you finish a short or long rest.

You "use" this feature when you "use your reaction to impose disadvantage"; at that point you have received a distinct benefit (disadvantage on the attack). What follows are further consequences of the "use" - you get advantage if they miss but not if they hit.

From my reading, no.

As you mention in the question:

This may come down to whether or not the Lucky roll counts as an "attack roll" itself or whether the "attack roll" was the original roll and the Lucky roll is some other type of roll.

Yes, this is the case. And Lucky is not an attack roll.

Lucky says:

Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20. [...] You choose which of the d20s is used for the attack roll, ability check, or saving throw.

You don't make a new attack roll - you just roll an extra die and then choose which die to use for the attack roll. The Lucky itself is not an attack roll, as much as attacking with advantage doesn't let you make two attack rolls or two saving throws, it is one attack roll that has a "more complicated" math than just rolling. Until you roll the additional d20 from Lucky, you are in the middle of the attack roll (just as you would be, for example, if you were rolling with advantage and rolled one die first, took a break and then rolled the other - you are in the middle of the same attack roll). Portent specifically asks "before the roll", so you can't use it.

For another example, a damage roll consists of all the damage dice summed, e.g. 2d6. Adding an extra 1d6 does not make an extra damage roll, just an extra damage die.