In case any are keeping score there is 6 different kokyu nage from morotedori in Saito "Takemusu Aiki" series of books. Also there are even other different ones in the "Traditional Aikido" series of books.

In practice there are shown several other methods not shown in either book.

Your Kokyu Nage at 3:24 is a good example of using Kokyu nage as a catch-all term in the Iwama system. I would now call that throw a maeotoshi henko, I always think of the classical, hip to hip throw as kokyu nage, like your kokyu-ho demonstrated in the first video. While it's not hard for a mid level student to understand that all throws using kokyu extension can be called kokyu nage, it can get very challenging for new students to get an idea what the throw is.