That is more or less the right set of criteria that I would go by in a "military-esque" definition of the term, (I am discounting all political bastardizations of the term of course.) but that does cover the aspects of what made the STG/MP43-44 series innovative and unique for their time.
The lines get very blurry because the term itself relies on other terms that are often not fully defined.
Individual weapon - easy enough, one man one kit.
Selective Fire - Semi, plus burst or automatic.
Intermediate-power cartridge - This one gets sticky, pistol is easy, but what does and does not constitute a standard rifle cartridge? Everyone will have a slightly different set of opinions. Not being obtuse, but most militaries consider the M16/C7 to be a Service Rifle and 5.56mm to be a standard rifle caliber now.
All these interpretations make it difficult to really pin down the definition.
Other aspects of the STG that were ground breaking for the platform but not covered above, the fitment of a pistol grip, the line of the bore is directly inline with the shooter's shoulder, optional non-sniper optical sights, no provision for a bayonet (most patterns) and medium overall length about 1 meter.
Take the FG42, has a number of similar features but not technically an "Assault Rifle" by the definition above due to caliber, is it a LMG? a Main Battle Rifle? etc.
Just garnering more discussion about the whole idea, FN FAL - fires a full-powered rifle cartridge, the 7.62x51mm. Follow me with this, 7.52 Nato was built up from the .308 Winchester cartridge (nearly identical) and .308 Winchester is a development evolution of the .300 Savage hunting caliber.
Now the .300 Savage caliber was designed to offer a cartridge that could approach the ballistics of the .30-06 Springfield, while at the same time using a smaller case that could be cycled through a short-action lever rifle.
Based on that a person could decide that the 7.62 Nato came about as downsizing of a .30-06 (full-powered) and there is is an intermediate cartridge, not a full powered/battle rifle round.
I don't truly classify 7.62 Nato as such personally, but it is just to illustrate that building a classification of a firearm on somewhat vague terms is bound to be filled with pitfalls.
Jumping to car terminology as a contrast, Porsche builds a sports car with only two seats and an engine behind the driver, and they call it the 550, we have the Porsche 550. Other companies build similar cars that have only two seats and engines behind the driver, are they considered Ferrari and Lamborghini 550's?
Of course not, but are they sports cars? Yes, but some cars that are sports cars do not have their engines in the rear, such as the Corvette, after that things start to get muddy again when trying to lump similar but unalike things into categories.
It's all rather silly, ramming things into categories is at best, a crutch of the layman to understand something in a very general fashion without truly learning about the topic.