There is no standard way of naming cartridges. The inventor can call it most anything they want. A .38 cartridge is roughly .38" outside diameter. The bullet that fits inside is .357". The "Special" is just a way to separate it from all other 38 cal chamberings.
Not quite, and marketing as far as the actual numeric name IMO had little to do with it. First, it was a common practice to name a caliber after the case caliber rather than bullet caliber. Of course, one might ask why, but it was and still is commonly done that way. Of course, in cases like the 357 magnum and 38 special, S&W introduced both, and yet named the one cartridge by case diameter and the other by bullet diameter. For the famous question "why is it 44 magnum when the bullet is .429?" it was because the parent case was 44 S&W special, which had a parent case of 44 S&W aka 44 Russian. So when you really think about it, based on the history of the predecessors of the 44 magnum, marketing played little role in the chosen cal designation "44". With the 44 Russian's development, it was simply a matter of practicality in regards to the goal of making a black powder round with enough ft:lbs to be a man stopper. Of course, there may have been other ideas at play, but giving a catridge a numeric name for marketing was not done as much as you might think, and IMO none of these examples qualify.
Interestingly enough, the whole marketing theory of the origin of "44" actually happened in reverse. What I mean is that the 44 cal EARNED its reputation LONG BEFORE such guns as a 44 magnum arrived. So rather than "44" because it sounds cool, or because thats what Dirty Harry had, it was actually because it was a very effective and for S&Ws, such as the New Century (aka their first N frame) the round put a little more steel (vs 45, which would make the chambers bigger) in the cylinder, thereby strengthening it. Also remember that the 45 Colt already existed, so S&W felt that leaving the 45 cal alone, at that time, was a prudent choice. Of course decades before then, there was the 45 S&W aka 45 schofield, which was not a major success, but a success in its own right.
The "special" in 38 special certainly has meaning. Remember that when that round came out, around 1898 (for the S&W model 1899 aka first model M&P), it was a different time. There was no magnum, unless it was wine you were discussing. In 1898, even the 45 acp was not (officially) around yet. At that time, words such as "special" (and later "super") denoted a significantly higher power ("magnum" didn't come out until 1935). Yea yea, I know "hey man, a 38 special is not very powerful" well that all depends on what you are comparing it to. I promise you its much more powerful than a 38 RF, 38 colt or 38 S&W. The cartridge was contrived by adding case length onto the 38 colt due to complaints about a lack of effectiveness with the 38 colt cartridge. Its kind of like how the 40 S&W came to be, in reverse. Also, if you ever bother to read about cartridges, there were other "specials" such as the 32 WS aka 32 Winchester Special and the 44 S&W special.