Order doesn’t matter. The emergency and secondary just need to be wide open. Main drain should have the gate valve.
Personally, unless you have a large tank (like 150 gallons or more) and need the extra flow, I don’t know if bean animal actually adds that much benefit over a single siphon and an emergency drain. I understand the extra redundancy, but having an external box and good weir, clogging two drains is unlikely. And in small setups without a lot of space, not having to plumb that third drain saves valuable real estate. In some of my bean animal overflow, I even just capped the third drain.
That said, I usually always just have all stand pipes and really only have flow going down the main drain. If you really want the tickle down the durso (secondary drain) just put it on one of the side drains and use the middle as main drain and emergency the other side drain.