You guys (except Habs) are making me crazy! Granted I didn't have far to go but still... Let's please get our terminology and expectations on the same page so I will be a little less confused when trying to figure out what's going on in your cookshack.
Generally when we talk about Brisket we're describing a cut of beef consisting of two components ("flat" and "point") separated by a ribbon of fat. Usually sold as a whole (and called a "packer") or divided into its parts (although points are generally hard to find). It's untreated, uncooked, and uncured - raw meat.
How you prep and cook a brisket doesn't stop it from being a brisket but it does change the terminology.
If you add seasoning and smoke, you have (wait for it...) Smoked Brisket. This, in various incarnations, is also called BBQ brisket, Texas-style brisket, etc. As an aside, IMO smoked point makes the best burnt ends and chili anywhere.
In Europe and many parts of the "newer world," brisket (usually the flat) is often made into pot roast or other stewed/braised dishes.
If instead you cure the brisket with a salt and (usually) pickling spice solution you've created Corned Beef which is typically sold in those plastic bags (not nearly as good as you can easily cure at home) and usually labeled "point," or "flat." If it's labeled "round" it isn't brisket but some other cut that's been cured/corned. In the UK, they usually corn a different cut of beef and call it "Silverside." You can then cook it any way you want - braise, boil, bake, smoke, or whatever. This is the basis for the "traditional" St. Patrick's Day menu (corned beef & cabbage plus too much beer).
To go a step further, if you start with Corned Beef (store-bought or homemade) and soak out some of the salt, let it sit in rub for a couple of days, and then cook it (usually by smoking and/or steaming) you have the ultimate sandwich meat - Pastrami. Again IMO, corned point (because of its texture and fat content) makes the best pastrami.
End of rant - my work here is done.