So, why ‘Special’?
“Special” is a cartridge designation when a new more powerful round is produced. The previous US military cartridge was the 38 Long Colt, which had proven to have insufficient stopping power in combat.
The 38 Long Colt had a muzzle energy of 195 ft-lbs, and the 38 Special had 270 ft-lbs with the same 150 grain bullet as the 38 Long Colt.
Dennis
Yeah, it was ‘special’ because it was a more powerful alternative to the existing .38 Long Colt cartridge.
A few years later there was a .38 Colt Special with a flat nose and a few years after that a .38 Special Super Police with a heavier bullet and lower FPS.
Over time all compatible .38 ammo just sort of became known collectively as .38 Special.
Other names for similar cartridges are Extra Long, Magnum, Improved and Super. Improved can also mean other things, especially in rifle cartridges, like changing the taper angle of the body or the neck to allow more powder or better gas flow.
There is a least one “Maximum” the .357 Remington Maximum, and what a beastie it is, packing double the energy of the 357 Magnum. I don’t know what they will name it if they lengthen that one - the “.357 Impossible”?
Incidentally, lengthening the 38 Long Colt into the 38 Special resulted in a ‘sweet spot’, where the best powder charges just fill the entire case, making for very small shot to shot velocity variations. It is an accurate cartridge.
The .308 rifle cartridge is a similar, very accurate cartridge compared to the 30-06 shooting the same bullet. In this case, the .308 is a smaller cartridge that results in a good case fill.
Dennis
Ignorance fought. I always thought the term “.38 Special” (although I knew “.38” was a caliber) referred to the small handgun ubiquitous in older cops-and-robber movies. I’m sure I still has its proponents as a strappable even today.
I found this nice article on the history. So it’s the “Colt Detective” officially. Colt Detective Special .38 Review :: Guns.com
Perhaps adding to your confusion is that Smith & Wesson marketed a slimmed-down short-barreled .38 revolver designed for plain-clothes officers. Its trade name was / is “Chief’s Special”: Smith & Wesson Model 36 - Wikipedia
As well there’s the longstanding term “Saturday night special” to refer to various cheap concealable guns of all (smallish) calibers and types favored by low-level hoodlums and miscreants of the era.
What? No it didn’t. “.38 Special” has only ever referred to one cartridge - the one linked to in the OP.
The cartridge is also known as .38 Smith & Wesson Special, which is particularly confusing because there’s also totally incompatible round known as .38 Smith & Wesson, AKA .38/200.
Why would one cartridge be “more accurate” than another? Isn’t rifle accuracy almost entirely in the barrel/stock?
No; there’s a host of factors at play including projectile size/weight, velocity, and energy as well.
I don’t know of the mechanics or physics but I’ve seen reference in reloading articles that best accuracy was achieved within a narrow range of pressures.
May have something to do with barrel harmonics or metal flexing under pressure.
In that a lighter bullet and a more powerful charge add up to more velocity and thus a flatter trajectory (generally speaking)?
Correct. 
ETA: much ninja’d but I’ll leave it here for whatever it may be worth.
Accuracy depends on all those things.
The bullets have to be consistent in weight & shape from round to round. As do the primers. As do the cartridge shape and volume. As does the powder load. And the packing of the powder. And the seating of the bullet in the case. And how energectically and at what speed the primer fires.
All the above are wholly within the cartridge system.
The comes the gun system:
How evenly does the firing pin fire the primer? How repeatably does the action seat the cartridge? How straight, smooth and radially symmetric is the barrel? How does the overall action contribute to, or detract from the gun moving as the trigger is worked past the firing point? how clear and stable are the sights versus the barrel alignment versus the grip/stock alignment? Depending on the action, how much does action motion after firing contribute to muzzle motion?
Some factors are much bigger influences than others.
A cartridge that ends up very close to full, or even slightly overfull with powder has the potential to eliminate some of the variables in how the charge ignites and combusts.
Besides, like auto nuts and their oil filters, it gives enthusiasts one more nano-important thing to macro-obsess over. 
“Special” is no longer a word that means anything. What’s so special about any FBI agent who ranks above Probationary? What’s so special about some 1953 Buicks? What’s so special about children who are severely retarded or disabled?
The definition of “special” in this context was given in post two of this thread.
Since the OP has been answered, please permit me a marginally related question:
What is the effective range of one of those little snub-nose revolvers? I can’t imagine it’s very accurate even across a modestly sized room.
Yes it did. The .38 Special linked in the OP had about a dozen variants over the years, all of which are correctly called the generic name “.38 Special” today, despite having many variations.
Another cite besides that in the OP:
This question has been answered at length by others, but I will just point out the real world evidence. The 30-06 cartridge in the M1 Garand was the target rifle of choice for quite awhile in high power rifle competition.
When the M14 replaced the M1, the new rifle quickly took over the high power competition. The M14 is quite similar to the M1 in its function, although it has a removable magazine that is easier to change during rapid fire than the M1’s 8 round clip. Now, to some extent this is because the M14 was the military’s official rifle, but the scores raised dramatically when the new cartridge was introduced.
The 30-06 and .308 can shoot the same .308 caliber bullet, and the 30-06 can achieve a bit higher velocity, but the .308 is just a more accurate round by a considerable margin.
By 1958, the first perfect score was achieved in national competition, 250-250v, in other words, 50 shots not only in the 5 ring, but in the smaller, tie-breaking “V” ring. After a few years of increasing ties, the targets were changed to incorporate a new, smaller tie breaking ring, the “X” ring. And the overall size of the black bullseye was cut in half. A shot in the X still scores a 10 and it is used only to break ties.
I will refer readers to this web page which discusses the exact nature of the increase in accuracy:
And a decade later, when the shooters got the hang of the new M16, that rifle and cartridge took over as the main combination.
I still prefer my M1, it is such a lovely rifle to shoot. The nice “Ka-thang” after each shot as the bolt cycles. The AR-15 (civilian, semi-auto version of the M16) just goes “pwing”.
And finally, I will throw in a plug for the sport of high power rifle competition. What an enjoyable sport it is. And contrary to all the long range shots in movies, we don’t use scopes. Just plain iron sights. 600 yards for the standard course, and out to 1000 yards for other events. High power competition still pays homage to the M1. The rapid fire stages still load 2, reload with 8. This was initially dictated by the M1’s 8 round clip.
Dennis
Missed the edit.
“With two rounds - Load!!!” “Actions remaining open except Garands”
Fiddle the goofy two round clip into the action and let the bolt slam home. Then wait in a crouch.
“When targets appear - commence firing”. Heartbeat. Heartbeat. “Commence firing!!!”
Drop in the prone position, plant the elbow, snuggle in to the sling and find the bull. Breathe. Sight picture. Breathe. Blam - thang! Again! Ka-chunk - the clip goes flying. Roll to the right, shove the full clip in and pull out. Watch the thumb!
Repeat eight more times. What a rush!
Dennis
Southern rock gone commercial, beats the shit out of disco … you kids today don’t know how bad it was in the late '70’s … so 38 Special was special because it weren’t New Wave and it weren’t Devo …