It's a tricky one. First off, ask your sensei, they know what they will expect of you on a test, and how your specific Dojo defines "Kokyu Nage".
However in the world of Aikido, Kokyu nage is a tricky one. Saito Sensei seems to use Kokyu nage as a catch all for odd ball throws. I have also heard it said that any throw done with "kokyu" can be called a Kokyu nage, however all Aikido throws use kokyu, so why have any other names? Well because that would get super hard to understand what we were talking about. At our dojo we define a "classic" Kokyu nage as the side by side throw where both Uke and Nage are facing the same direction, like this http://www.aikidostudent.com/oldasc/content/?p=238
Every Aikido throw is kind of like this, I think Ueshiba was pointing at principles when he named his techniques, and not the mechanical specifics of the throw. Irimi Nage, can be done many different ways, in fact one variation of Irimi nage is the exact throw I showed above as Kokyu nage- it can be called an Irimi nage as well though, because you must enter with your body to do that throw. When it comes to naming the Nage waza of Aikido you have to get a feeling of what part is important in the variation you are doing, then it's clear as to why a teacher would call one throw something (like irimi, kokyu, kaiten) instead of calling it something else, the main principle of the throw is the important part, not the mechanical specifics. This all makes understanding Aikido nomenclature very difficult for the newer student.
Hope that helps!
When I teach I always refer to techniques as "classic" when I say that my students know that I'm saying this is the mechanical specifics of the throw most commonly called by whatever name I'm using. Some times I will even say something like this is a "classic Iwama" or "classic Aikikai" etc, so they know that one lineage uses that name to describe these mechanical specifics. So again, ask your Sensei, they know what they want to see on tests, and can give you their interpretation of why a throw should be given a specific name.