Like what YinYoru pointed on on Pointer 2 you brought up.

Passive Barriers are not determined by the Ninja Position, instead Passive Barriers are determined by the Team's total initiative, be it whatever combat system you're using (i.e. Ranked, PvE, PvP, Space-Time, etc).

Active Barriers are however determined by the Ninja Position, and if in the same position, the one with higher initiative will activate the Active Barrier first (though some argue the higher initiative 2nd, but I always see the higher initiative activate first, and this observation of being the higher initiative isn't from counting the initiative points or whatnot from opening their equipments and counting. Instead it was based on a normal standard attack of going first against the opposite mirrored position). But this initiative for Active Barriers, I can't be 100% certain who goes first, but its definitely initiative dependent.