My conure is extremely intelligent at times, learning a trick at the second try or doing what I ask him immediately. Most of the time, though, he understands but decides to just ignore me.
What do you mean by this? Surely this applies to humans too, we are animals after all. So what distinction did you intend to make?
With humans, performance in one cognitive test correlates with another and so on, generally. So, intelligence across domains.
Researchers test the same with animals. The issue being animals' intelligence being tied to their ecology. The dilemma being what is it worth for an animal solving a task that has no significance in its life. The other argument being if the animals' intelligence is closer/similar to human intelligence, we will find similar results in both.