First Gendo is right, this isn't an activated ability but a triggered one (I do think this was more a mistake in the wording you used, rather than a lack of understanding on the abilities, particularly since you had the right word in the question, so I won't address this part any more.
I think you may be thinking about this wrong, It sounds like you're thinking it only applies to your turn, or that it might be working on some idea of per round, if this was how Jolrael worked, her ability would be written "whenever you draw your second card on each of your turns, create a 2/2 green Cat creature token."
Before you played Jolrael on your turn, you had drawn your first card. If you had drawn another card on your turn, for any reason, Jolrael would have seen that second card drawn and give you a cat token, but you ended your turn without drawing a card, so your cards drawn this turn is back to 0 when your opponent's turn starts. Your opponent cast a spell that makes you draw 3 cards. Cards in magic are drawn one at a time (Rule 121.2), so you will draw the first one, your first that turn then the spell continues and you draw your second card for the current turn. The game sees this happen, but does nothing about it yet, this is covered in the Comprehensive Rules for triggered abilities (emphasis mine):
603.2 Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.3 Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.
So with these rules in mind, the game recognizes Jolrael's condition was met, but Burning Inquiry isn't finished resolving. Everyone draws their third card then randomly discards three from their hand. This discard doesn't change the fact that 3 cards were drawn. At this point your opponent, the Active Player, will get priority since it is his turn, but this is the time the game puts Jolrael's ability on the stack per rule 603.3 and give you a cat token when it resolves.
Players are more likely to draw cards on their own turn, particularly when drawing more than one, since single draws on your own turn become the second card after your draw for turn, but all that Jolrael cares about is that you drew card number two for the current turn.