I've had parrots since I was a child and bred green cheeked conures through undergrad. One of my major objectives has been to intervene during early development to establish human-avian communication via operant conditioning. This is a fancy way of saying that I wanted to teach them to do tricks at an early age so that they knew how to navigate a human world with simple communicative symbols.
The above image (copyrighted by me) is my depiction, based on my life's experience, networking, and the few journal articles available, of where popular species stand on these axes. This was inspired by Arthur (my un-cape parrot) and several other species with whom I've worked to develop trick training. While my conures, whom I brought to class, are fantastic workers, my Arthur is not. This is the bane of several un-cape owners' existence, especially since they are notorious for their high emotionality and intelligence. It occurred to me (as well as a few others) that some species may be "too" smart for formal training, or at least food-based training. I made this chart because it is the first of its kind, and it seems to be a decent representation.
On the far right, you have high cognitive complexity (very loosely defined since this is a rough draft). Cockatoos, macaws, Greys, and other difficult species (including a crow for reference) make up the majority of this hemisphere. However, while cockatoos on the whole are much less likely to respond to food-based reward, macaws are famous for their trainability. On the far left side, you see an interesting cluster in the top pole. Although all parrot species do qualify as cognitively complex (as are most vocal-learning types of social animals) in reference to, say, reptiles, they are not known for being emotional circuses like Cacatua.
It seems that there is a "just right" combination for trick training, if you also want a less cognitively complex trainee, and that interactions may occur as complexity progresses across the chart. In essence, the more complex you are, the more room for differences in trainability.
You'll notice that Greys are on the far right and in the middle. I put them there because they are very good at training, but shy away in front of strangers.
If you have an individual who seems to defy this trend, that is interesting. However, keep in mind that individuals are just that- individual. Nonetheless, there is much to be learned, and as the future continues I am excited to incorporate additional axes and measures.
Please do not use the chart without first acquiring my written consent :)