"9mm Luger" is about as specific as saying there was a 12oz soda can on his table. The details about a specific cartridge are only written on the box, but can be determined from a casing and a photo book. If he said he has "124 Gr Remmington Golden Saber 9mm Luger" ammo, that would actually be specific information.
(Gun illiterate).
Today, "9mm Luger", "9mm Parabellum", and "9x19mm" are all synonyms for this same cartridge. Usually when "9mm" is said without qualification, this is what it means. However, there are other much rarer 9mm cartridges such as the 9x18mm Makarov, and also "9mm short" and "9x17mm" are rarely-used synonyms for the (common) cartridge usually called .380ACP.
There is a small difference in width but you can't really tell if you're further than a foot away.
Edit: Also, there is usually an imprint on the bottom of the cartridge that says "9MM LUGER".
You're right that there are imprints on all rounds but seeing and remembering that is like getting a database error and remembering if it was Microsoft or MYSQL...You have to have a lot of experience and a keen memory.
So pretty much impossible.
[1] http://www.wired.com/wired/archive/11.06/battlefield_pr.html
Edit: spelling
https://en.wikipedia.org/wiki/File:CartridgeComparison.jpg
https://en.wikipedia.org/wiki/File:Comparitive_handgun_round...