A:
9mm Parabellum, 9mm Luger, and 9X19mm all mean the same thing. This is the most popular round in 9mm size, and so when someone just says "9mm", then they are referring to this chambering. You will find many examples of 9mm diameter cartridges, and they typically vary based on the lenght of the bullet. The diameter remains fairly constant.
Another example is the .380 ACP round (9x17mm) which is known is Europe as the 9mm Kurtz (short). Other names you find are 9mm Largo (9x22) and 9mm Makarov (9x18).
Yet another example of a round using the 9mm diameter round .38 special. The 9mm diameter is actually .355 inches. English naming typically uses inch measurements, whereas Europe and others use metric. But some liberties are taken with the exact number used in the name.
You can go on to find a lot of examples, but I'll end with this...9mm Luger, .38 Special, and .357 magnum all have the same diameter. And each are offered in a 115 gr weight bullet round. I'll let you guess which is most powerful of the three.