You ask a very good and reasonable question: It can be confusing. The problem is that when they name cartridges, especially in "the old days," they (the manufacturers) could be a bit less than accurate.
First off, there's often a slight difference in bullet diameter in the same caliber between cast and jacketed bullets. For instance, a 9mm should work out to .360". However, depending on the actual bullet used, it will be either .355" or .356".
In the case of the .38 special, it is actually a .357" bore. Don't ask why: I don't know. But it was an "improved" version of an earlier cartridge, as is the .357 Magnum. The key difference between the .38 special and .357 Magnum is that you've got a longer, stronger case, allowing for more powder and a bigger, heavier bullet. The handgun is also going to be larger and heavier, because that extra power means higher pressures. Here's a photo I found of 4 cartridges with roughly the same bullet diameter: The only "interchangable" cartridges are the .38 special and .357; And you can only fire the .38 special in the .357 handgun, NOT the other way around. The .380 is sometimes known as the 9mm short, and it fires the same diameter bullet as the 9mm, .355" or .356".
Another example of "bad naming" is the .44 Magnum. It uses a bullet diameter of .429". And as with the .38 special and .357 magnum, you can fire lighter cartridges through it, like the .44 Russian and .44 special. Interestingly enough, they got the name right on the .41 Magnum: It's got a bore diameter of .410".
Mark