Almost got it covered
Caliber designations (round names) may or may not be the bore diameter, groove diameter, bullet diameter, or they may be something else entirely.
If the manufacturer thinks it is a good name, then there it goes. Generally round have either a name close to their actual measurement, or close to what they once were.
US .38 caliber revolvers fire a .357/.358 in dia bullet. But once they used an actual .38 cal. The same for the .44, which are actually .429. Close to .43, but does .43 Magnum have as much appeal?
Names are chosen 1) close to caliber, and 2) so they won't be confused with another cartridge. .22 caliber centerfires have been named from .218 to .224-something. And then there are others.
Some rounds name includes the year of US military adoption (.30-06), others include the original parent case (.22-250 Remington, a wildcat .22 developed from the .250 Savage case, and years later standardized by Remington).
Old blackpowder rounds once had the caliber, powder charge, and bullet weight as their name, along with the maker (.45-70-405 Govt), which was shortened to just caliber and powder charge (.45-70), and this trend was carried through a few of the first smokeless powder rounds (.30-30 Win, .30-40 Krag).
There are lots more quirks, and metric calibers follow a different set of rules, generally being caliber x case length, in mm (7x57mm Mauser), adding the letter R for a rimmed case (7.62x54R)
These are general, and many exceptions exist. A great reference on the subject is Cartridges of the World (Barnes). Reloading manuals also usually contain a short history of the different rounds.
Hope this helps.