9mm Luger is the most common modern definition for the 9x19 cartridge; Parabellum name is what it was called early on and still referred as that today by some. About the only distinction that bears merit is the 9mm NATO designation for the 9x19 cartridge; as long as your firearm is rated for what is called '+P' pressures you are good, as the NATO loading is a hotter or faster load causing more stress on your firearm which could be detrimental over prolonged use. 9mm Short, Kurtz, Makorov , 9x21, 9x23 are all very different cartridges that CAN NOT be used in a 9mm Luger/Parabellum (9x19) firearm.

different name for the exact same round the 9mm NATO is slight hotter

This web site gives the best information on 9mm that I have found: ammunitiontogo.com/lodge/9mm-luger-ammunition-names/ I will add if you are reloading and the brass head stamp has a nato mark you will need to remove the crimp in the primer pocket before you try to put the new primer in.

A chart of cartridge sizes is available on WIKIPEDIA . This is the link https://en.wikipedia.org/wiki/9_mm_caliber

9mm Lugar, 9mm Parabellum and 9x19 are all used to describe a dimensionally identical cartridge. When the cartridge was introduced It was called 9mm Parabellum to differentiate it from other 9mm cartridges at the time. 9mm Lugar tends to be a US designation. U.S. loaded ball or practice ammo tends to be loaded a bit lighter ( lower velocity) than european ammo, unless the U.S. ammo is marked +P, +P+, or NATO. US, european or NATO ammo should be fine in any modern (1960's or later) firearm. If you have a european gun it may be more accurate with NATO or european ammo because the sights are regulated at the factory for the local ammo. All new 9mm pistols sold in the US are good for NATO ammo, but check to see if the gun is rated for +P or +P+ before using. I would avoid military surplus 9mm since some is loaded VERY hot for submachine gun use.

9mm Luger is called by several names. But it is still the same round. Officially, it is marketed as 9mm Luger. Use them identically to each other.
“Parabellum” is simply a Latin-based term that means to prepare for war. DWM, the company that created the cartridge, held the motto, “Si Vis Pacem, Para Bellum” which means “if you seek peace, prepare for war.”
So, these all mean the same thing:
9mm Ammo
9×19 Ammo
9mm Luger Ammo
9mm Parabellum
The 9mm you do NOT want to mix is the Makarov. It is 9x18 instead of 9x19.

Do more research but the dimensions of the two are the same and to make things even more interesting, the 9x19 and 9mm nato have same dimensions. Without going through the history of 9mm, 9x19, 9mm Luger and 9mm parabellum, i have a simple solution to your question. To cut down on confusion and possible damage to your mass accelerator, ALWAYS USE THE DESIGNATED CALIBER MARKED ON THE MASS ACCELERATOR being used. It is that simple. If it is an older weapon and worried about possible damage to over pressure, then please do your homework and read up on the the gun you are going to use. MOST modern day firearms will fire all four designations of this round. Not a simple answer for such a short question.

The two cartridges are interchangeable but not all 9mm ammunition is created the same. There are several different organizations that oversee ammunition standards. Here in the U.S. the Sporting Arms Ammunition Manufacturer’s Institute sets standards for civilian ammunition. There are European, Asian and other standards as well as various military standards.
The problem lies in the fact that there are a number of older guns that can chamber the original 9mm Parabellum round and then disintegrate upon firing. The Italian Glisenti from Word War 2 is one example. For U.S. civilian ammunition, SAAMI set 9mm Luger pressures artificially low to prevent this. +P loads were developed to more closely duplicate the original loading. Do NOT use +P loads in any gun until you are sure that it is capable of withstanding such loads.
The original 9mm Parabellum loads were loaded to very high pressures; on the order of .357 Magnum pressures. Modern military loads that duplicate these loads are often found in gun stores marked as 9mm NATO. Do NOT use such loads in anything but a modern military gun capable of using such ammunition.
DO NOT fire any 9mm Ammo other than is designated (stamped) on the firearm !!!

Same round! You may find ammo that is labeled for "Submachine Gun Use Only". So you are at risk if you use that ammo in most modern handguns!

It is the alter ego of the 9x19, they are seeing a Shrink to sort this out.

They are essentially the same. The Luger should not be used with high pressure loads, but it was some what problematic with ammo. Note they did have hot loads but were for the sub machine gun.

9 mm Luger and 9 mm Parabellum are the same for hotter loads they have 9 mm +P and +P+make sure your firearm is rated for them

I think they are basically the same, the Luger may work the toggle on the German guns better because it is hotter, but that is the only difference.

The Luger pistol was named after its designer, Georg Luger. The 9mm parabellum cartridge is also called simply “9mm” because of its ubiquity and “9mmx19mm”. “parabellum” comes from the Latin phrase meaning “essential to prepare for war.”

No, that’s not true. At least not now. SAMI used the name “9mm Luger” in their cartridge designation because of its history but they claim the cartridges to be the same.

The Luger is a specific handgun, while "parabellum" is the name of a cartridge. That's technically true, but 9mm Parabellum and 9mm Luger are really just the same thing. There are other 9mm cartridges which are different. For instance the .380ACP is sometimes called a 9mm Kurtz (meaning "short")

No Luger and Parabellum are just different names for the same thing.