- the_shadock
I just got the answer, and it's quite complicated...
Nah, it is not that bad compared with the designation mess of some other vehicles, particularily the whole Jagdpanzer/Sturmgeschütz/Panzerjäger debacle that sprung from the bureacratic infighting between the Generalinspectorates of Artillery and that of the Panzertruppe 
- the_shadock
The official denomination for the StuG III with the 7,5 cm long barrel was first StuG 40, because it carry the gun SturmKanone (StuK) 40,
<SNIP>
That is right for the models Ausf. F, F/8 and G. The models Ausf. A to E (7,5 cm short barrel) were then called StuG III.
You can find StuG 40 Ausf. G with or without the Saukopf, it doesn't matters.
But when appeared the Sturmgeschütz IV (based on a PzKpfw IV hull), all the StuG 40 were renamed StuG III, so that they could easily be differentiated from the new StuG IV. So, you can find the same denomination for different vehicles, which makes it a bit complicated.
You also have a variant with the 10.5 cm Howitzer, which is called 10,5 cm Sturmhaubitze or StuH 42.
I'm not sure what your source is for this, but to some extent is clashes with what Jentz has to say in his Osprey volumes on this vehicle. As far as I can make out, the story goes like this:
Officially, all StuGs until some point in 1943 were designated "gepanzerte Selbstfahrlafette für Sturmgeschütz" (armoured selfpropelled chassis for assault gun). This was then qualified by adding the designation of the weapon it carried and the mark of the vehicle, so an Ausf. E with the short 7,5cm StuK 37 would be :
"gepanzerte Selbstfahrlafette für Sturmgeschütz 7,5cm Kanone Ausf. E"
When the long 7,5cm gun entered the picture early in 1942, the official designation was:
"gepanzerte Selbstfahrlafette für Sturmgeschütz 7,5cm Sturmkanone 40 Ausf. F"
My guess would be that the StuG 40 designation came about when some beancounter needed a way to discern between the abbreviation previously used for the vehicle armed with the short-barreled 7,5cm - "StuG" - and the new ones armed with the longer 7,5cm gun and so coined the phrase "StuG 40".
Something similar appears to have happened in the case of the Panzer III, where a Panzer III was just PzIII to begin with, then they became PzIII(37) and PzIII(5) when the short 5cm gun arrived, then PzIII(5kz) and PzIII(5lg) (kz = kurz= short, lg=lang=long) and finally PzIII(75) when the Panzer III Ausf. N arrived with the short 7,5cm gun.
Anyway, in late 1942, the 10,5cm Sturmhaubitze was mounted in the StuG but I haven't found any early designation for that vehicle, other than "Sturmgeschütz leichte Feldhaubitze" that was used in the project phase back in 1941.
Now, early in 1943, enter the vehicle affectionately known as the "Brummbär", a Panzer IV with a tall, box-like superstructure mounting a short 15cm howitzer. Now there were StuG-type vehicles based on both the Panzer III and Panzer IV and someone apparently decided to clean up the designations. So, the vehicle previously known as
"gepanzerte Selbstfahrlafette für Sturmgeschütz 7,5cm Sturmkanone 40 Ausf. F" (or F/8 or G)
became simply
"Sturmgeschütz III für 7,5cm StuK 40 L/48"
The howitzer armed vehicle became
"Sturmgeschütz III für 10,5cm StuHaub 42"
while the Panzer IV based "Brummbär" became the
"Sturmgeschütz IV für 15cm StuHaub 43"
When the Alkett factory was bombed in late 1943 and StuG production ground to a halt, StuG superstructures were mounted on Panzer IV hulls at the Krupp factory, producing the
"Sturmgeschütz IV für 7,5cm Stuk 40 L/48"
It would seem that by 1944, that got a tad too confusing as well, the "Sturmgeschütz III für 10,5cm StuHaub 42" becomming the "Sturmhaubitze" and the "Sturmgeschütz IV für 15cm StuHaub 43" becomming the "Sturmpanzer", but I cannot follow that debacle in my sources.
The crux of the matter is, as with all German AFV designations, that they are dynamic and subject to change. New designations will be created and retrofitted to older designs, some agencies will coin their own designations suiting their particular purpose (tabulating strength returns, for example) or bureaucratic powerstruggles will expand into the area of naming and concepts. Furthermore, perhaps not everybody will get the message and change the designations in their correspondance until long after it has officially changed.
If I was you, I'd simply forget about official designations and use something we can all understand and follow
Claus B
