A few points to discuss...
I have heard a slightly different story about the Sturmgewehr, and sadly, can provide no references at this time, so consider it anecdoatal, but entirely plausible.
During the years when the Nazis were winning, developement of many weapons were either given a low priority or canclled entirely. Hitler had ordered further developement of submachineguns (MaschinenPistole) be suspended, as he felt there was no need for wasting the resources. He did, however allow further developement of rifles.
Without Hitler's knowledge the MP 43 was developed (this in not a unique instance, several different designs of weapons, including some aircraft were developed without "official" approval, and sometimes inspite of official orders), and small numbers were sent to the Eastern Front for troop trials.
At some time a few months later, at a conference with some officers returning from the front, Hilter asked what they needed, and they told him "more of the new rifles!" Hitler was perplexed, and on learning the details, enraged at having his orders flouted. However, shortly afterwards, he was given a demonstration of the "new rifles" and became an enthusiastic convert, nameing the "Sturmgewehr" (Assault Rifle). The designation was then changed from MP44 to Stg44 (from machine pistol to assault rifle).
Captured examples led the Allies to form the general definition of "assault rifle", based on the main characteristics of the Stg 44/45. Selective fire, intermediate cartridge, were the defining characteristics. Box magazine fed, straight stock, pistol grip, these were all common features, but were not what was used to define "assault rifle".
The standard of "intermediate power" cartridge was based on the common rifle and pistol cartridges in use during WWII. SO, anything more powerful than the standard pistol rounds (9mm, .45acp 7.62x25mm, etc.) and less powerful then the standard infantry rifle rounds (.30-06, 8mm Mauser, 6.5/7.7mm Jap, 7.62x54R, .303 British, etc.) was considered to be "intemediate power. And yes, this did include the .30 carbine, as it was above "regular" pistol power. Later on, the 5.56mm (.223 Rem) was classed in the intermediate power range, due primarily to its small bore size.
Oh, but there are semi-automatic transmissions. And now here's another history lesson.
I've certainly heard the story of the MP-44. There was also a different rifle called (I think) the Volksturm, which was also chambered for the same short cartridge. It was semi-auto only and even more importantly had no pistol grip, so it could not possibly be an assault rifle, although it had a high capacity magazine--so maybe it was. Parts of it were black, too. At about the same time, there was also developed for air force paratroopers the FG-42, which was select-fire, had a "high capacity" magazine and even a pistol grip, yet was chambered in the standard 8mm rifle caliber. So was it an assault rifle?
The Volksturm(a last ditch effort fielded in small numbers in 1945) would not be an assault rifle, if it lacked the select fire capacity.
The FG-42 (adopted in 1942) was a marvel of engineering, firing semi auto from the closed bolt, but full auto from the open bolt. But it does not fit the assault rifle definition, because it fires the full size 8x57mm infantry cartridge. I would be considered a "battle rifle" by the definitions we use today.
Even earlier, there was another infantry rifle chambered in .30-06 that was select fire, though it had no bipod or pistol grip but it had a high capacity magazine and was select fire. The marketing people claimed that it was intended to be used in the assault. You were supposed to fire a burst when your left (or was it the right) foot hit the ground. Was it an assault rifle or not? It was designed by some guy named Browning but I don't know whatever became of the idea. It sounds like it had promise.
Promise it did have, and it delivered! The BAR was considered an automatic rifle, (hence the name), but is technically a light machine gun, and some versions were full auto only, and had bipods. Introduced along with the BAR was the concept of "walking fire" and there was even a special bracket for the belt to put the butt of the gun in when advancing. Remember, this was 1918, and many concepts that looked good on paper didn't turn out quite as well in the field.
If you chambered something in 6.5 Jap or 6.5 Swede, could you make an assault rifle? Wasn't the 7.62 NATO considered an intermediate cartridge? Wasn't the M14 an assault rifle? I saw them with pistol grips when I was in the army and you can get them in black now, for the purist.
6.5mm Jap & 6.5 Swede would not be proper assault rifles, as they were considered "full power" infantry cartridges (although on the light end of the accepted power range in those days).
The 7.62 NATO was never considered an intermediate power cartridge, as it has the FULL POWER of the GI .30-06 load. Its 1/2" shorter length over the '06 has benefits for rifle design, and a slight saving in weight over the '06, but it is by no means an "intermediate power" round.
Where it gets really confusing to those who have not studied it is the translation of the word
Sturm into English, and the language used by the anti gunners in crafting the 1994 AWB.
Sturm is generally translated into English in two ways. First is "storm", and meaning either
storm as in weather, or to
storm an objective, in the military sense. Which storm it is, depends on context. The other common translation is assault, which again refers to the military assaulting an objective. Confusion arises when people use the other English defintion of assault, the one used in law, of one person attacking another. In the broadest possible terms, it is correct, since both assault a bunker and assault an opponent involve an attack intended to cause harm, but using the English defintion of assault on an individual is ...imprecise, when translating the German term
Sturm.
Fast forward to the late 80s and early 90s, and the anti gunner's media
assault 
on military look alike firearms ownership. Mass murders by crazed gunmen (the Stockton schoolyard shooting was the beginning of the medi blitz, focusing on the {semi auto} AK 47 used there. Other incidents, including the copycat shooting at a Kentucky printing plant, each using a military looking rifle, added further "ammo" to the media effort to prove how evil and dangerous thsese guns were, convieniently and constantly ignoring the fact that the men doing the shooting were mentally unbalanced, and nearly all of them on prescription mind altering medication!
In the beginning, they tried calling the rifles assault rifles, but we (and some others who were, unlike the media, concerned with facts), pointed out that they were NOT assault rifles, but semi auto military lookalikes, no different mechanically than many models of common sporting arms.
The media then tried for a while to call them "semiautomatic assault rifles", which was, I believe, their idea of being accurate. This phrase "semiautomatic assault rifle" soon prooved to be too cumbersome for a good sound bite. SO they settled on another term, one they made up, "assault weapon". This had the benefit, for them, of being a convient sound bite, sounding very much like assault rifle, and (even more importantly) since it was their made up term, they got to define it.
And, define it, they did. Their definition of an "assault weapon" did not include machine guns, OR and actual assault rifles! (which, under US law are machine guns). They defined asasault weapon as any semiautomatic firearm (rifle pistol or shotgun) with a "magic number" of certain features. Sometimes it was two (most usual), other times the magic number was three of the "evil" features. And those features were such evil things as:detatchable magazine; pistol grip; bayonet mount(lug); flash suppressor/flash hider, forward handgrip; heat shield; folding stock; any handgun NOT taking the magazine through the pistol grip, etc.
This defintion became codified into US law with the 1994 Assault Weapons Bill (AWB), and several state laws that basically mirrored the Federal one, without the Federal law's sunset provision.
SO, assault weapons are semi auto firearms with two (or more) comsmetic features from their proscribed list. NO single
assault rifle was affected by the 1994 AWB, only semiautos.
And while the Federal AWB sunset in 2004, several state laws covering essentially the same ground, did not.
Now, with people's usage of language being as sloppy as it is, the difference in common conversation between Assault weapon and assault rifle has become even further blurred, with most people not recognizing, or caring about the difference. TO many people, if it looks like a modern military rifle, or has a large magazine, its an assault rifle or assault weapon, assault pistol, assault shotgun, etc. And that was the intent of the anti's when they created the term assault weapon.
Even a great many firearms enthusiasts use the terms incorrectly, usually without knowing they are even doing so.
