I dont know man, theres so many books that say different things. I always thought that all assault guns, Marders, Hetzers, Jagdtigers, Jagdpanthers, stugs, Jagdpanzers, Brumbars, and sturmtigers could "technically" fall under the umbrella of the term Sturmgeshutz.
I wouldnt think that an infantry unit that was being reinforced for example would be told whether they were being reinforced with assault gun type A or assault gun type B. I would think they'd be told "A platoon of assault guns will be joining us."
Again Im no expert, just an enthusiast.