Question about the Weapon Banner

Nope, iirc after you pull the banner weapon (either the one on the path or the other) the count reset back to the beginning.

No, for weapon banners the only thing you keep is the pity count

Short answer: no. Long answer: Once you lose to a standard weapon the VERY NEXT 5\* pull is guaranteed to be one of the limited ones. Since after getting the standard weapon you continued to pull until getting Flins' weapon (which IS one of the limited weapons, so the wording still applies) your odds reset to normal after that.

You would've been guaranteed one of the banner weapons if you either reset your fate point for whatever reason and kept pulling on the same banner (so 50% chance of getting Flins' weapon or 50% for Yelan's) or hadn't continued pulling on the banner until it changed to another one. As it is, you got your chosen weapon through the epitomized path mechanic. The guarantee after losing to a standard weapon is a bit redundant now since they reduced fate points from 2 to 1 and most people will likely pull until they get their desired weapon, but if you don't cash in your fate point guarantee after getting a standard weapon, you'll be guaranteed a limited one the next time you pull a 5\*.

The first two comments here are wrong. If you lost to a standard weapon, your next weapon is guaranteed to be one of the limited ones. In this case, it's either Columbina or Ineffa's weapon.