But based on the FAQ I can't see any point or scenario where the Sun's light would not form some kind of conical frustum.
I'll take this slowly... trying to help. Forgive any inferred insult.
1) FAQ says the Sun moves along a circle over the Equator on the equinox.
2) On the Equinox, though Tom Bishop argues against this, the day last exactly 12 hours. (Well, the poles don't, but forget this expection for now.)
3) To make this work, the Sun must illuminate 1/4 around the Earth from pole to pole in both directions and no farther.
4) At any given time, half the Earth is illuminated only.