I'm a little unsure of how the Weekly Reset works, here is how I think it works currently, please correct me if I'm way off.

1. You kill the Primals in any order after your first attempt for the weekly loot, but for your first attempt for each fight, you have to do Garuda, then Titan, then Ifrit.

2. You will get one weapon after finishing all three, you are counted as having progress for Garuda if you killed her last week, then Titan the next, so long as it's your first time at getting the achievements. Otherwise, your progress will reset each week.
Or do you have progress for them until they reset weekly but you will still only be able to receive one weapon for the week when you finish Ifrit even if Garuda and Titan were killed in a previous week?

I really don't know what the case might be since I just finished Titan for the first time and turned in the quest, can someone fill me in?