Someone once told me to take calipers and measure the dimensions of the loaded case of a 38 and 44, seems that is where the dimensions for the naming of the cartridge came from more than anything to do with the actual bullet diameter itself.
38Special once had a more potent familiar relative, the 38/44 cartridge. Was essentially a hot rodded 38Special that was built on the heavy duty frame of the 44 Special(N-frame).
Problem was that 38/44 ammo could be loaded into smaller framed guns meant for ordinary 38Special ammo, care had to be taken with such rounds to not get them mixed up. Much like using 38+P of today in guns not rated for them.
To solve the problem, the 357 Magnum was created by lengthening the 38Special case .1 inch. It's enough to prevent chambering 357Magnum into a 38Special. The name 357Magnum was likely to help people differentiate between the two cartridges, rather than have a "38 Magnum" they went with bullet diameter when making the name.
357Magnum is a relative of the 38 Special, case is just lengthened hence the reason why 38Special can be fired in guns chambered for 357Magnum.
Similar relationships as others have noted between 44Special and 44Magnum, 45LC and 454 Casull(though there are some dangers associated with doing it often in the 454Casull).
Btw, here's a picture of the 38/44 Outdoorsman sitting in my collection. I've hotloaded for it on a few occasions with 158grn LSWCs at around 1200fps, a touch over what you see most +P loadings for 38Special loaded to these days.