What's interesting is, depending on the year you look at, some years it seems the second budget email doesn't even matter? That's from looking at the spreadsheet shared a while back in this thread and past threads, where there have been instances of people not receiving a second email and getting the award. Then again some years it seems to be an unmistakable correlation ¯\_(ツ)_/¯
Re: flagship applicants, the second budget email I got was from an address called "BorenAFLI". My budget was definitely not questionable, as I'm only intending to participate in the official programs paid directly by Boren. Still got a second email so my guess is that every FLI applicant gets the second one--which seems consistent with this thread. You're also right that FLI applicants have better chances, it seems Boren has money set aside for these programs, aka a set amount of spots it needs to fill every year. According to this report from 2017 the FLI acceptance rates tend to be around 50% https://nsep.gov/sites/default/files/NSEP Annual Report 2017 FINAL with Front and Back Covers.pdf