The person in the picture looks like Hitohiro Saito Sensei but I could be wrong. I agree the name is a label it really doesn't matter what you call it, it is what it is. However in Iwama Ryu, this technique is called Morote Dori Kokyu ho, again you are right it is an exercise practised at the start of every class in every Iwama Style Dojo. From any other attack it is kokyu nage. It is a 'gateway' technique.
Itīs actually kokyu-ho from a lot of other attacks than morotedori but it can also be a kokyunage. If done with a direct entry (with hitoemi, as in sixth and seventh suburi) itīs very like what others call sokomen iriminage.