Yes, an adjacent creature must beat your armor class to suffer Corona damage.
The question of whether you play Starjammer (abilities) or Starfinder (tags), or a hybrid of the two, is not clear, but the answer is the same in all cases.
If your result is equal to or greater than the target's armor class, you deal damage.
Your armor class is the minimum result of an attack roll that an opponent needs to hit you and inflict damage.
Conversely (although this is not clearly stated), if an attack roll is not equal to or greater than the target's armor class, it does not strike and cause no damage. According to the rules, it is only one type of "hit". If there was another, there would be a rule and a name for it.
Sometimes the GM can tell that a "missed" attack has bounced on your armor, but the rules do not distinguish between that and any other type of failure. It still would not trigger the crown.
The electrostatic field behaves in the same way, despite the difference in formulation. I would say that it is an inconsistent writing and not a difference in the designer's intent.
The order of operations on the damage is not specified by the rules, but at my table, I would say that both occur as independent events that do not affect. They still suffer the damage even if their attack closes your Corona in any way and you still suffer the damage from the triggering attack even if your Corona kills them or prevents them from taking further attacks.