What model M4 do you use the sight with? I've been in a small debate with a teamate on what classifies a rifle as a DMR over a BR, with me giving your definition of a DMR. The rifle the sight will be mounted on fits the definition.
At the moment I use a CA sportline M4 with a carbine stock, MOE grip and MOE hand-guard, installed with JG S-System fold-down sights. But I do plan to change the outer barrel to a 10.5" CQB-R length one. Also I only ALMOST use this setup, as I am using a Hurricane 552 style EOTech replica
I also have the luxury of owning a custom built Yugoslavian Zastava M76 S.A.S.S.(400 fps, hard semi only) so I can also use this setup on that (if we are not playing on an official game).
In my definition a DMR is a rifle that is ALMOST similar to the one your average grunt is holding next to you on the outside. The main reason behind this is painstakingly obvious in the battlefields of the 21st century (and in airsoft too): if you stand out from your unit you get shot. Also a DM is not some kind of an uber-shot super ninja (like a Sniper), but more like a normal Marine (meaning he knows how to hit targets at long ranges, and he is a better shot than the average grunts). So the rifle he will use must be roughly the same as the one his teammates carry, but he will definitely use a setup that lets him fire to longer distances easily.
Forgot to answer the second part of the second question: A BR is a Battle-Rifle. A battle rifle is a selective fire platform, that fires a full size ammunition instead of a carbine sized one. So an M14 variant that was not rebuilt as a DMR platform would count as a BR. The Yugoslavian M77 definitely counts as a BR, as it is basically an AK chambered for the 8mm Mauser or 7.62 NATO round. We have to keep in mind, that a Battle rifle was not intended to give accurate fire to long distances, as it's actual goal is to give a greater stopping power at the distances your grunts are shooting at. Sometimes it is used to shoot through walls and covers or to take out targets in advanced body armor. A H&K G3 should also be classified as a BR.
To sum it up: Both differ from the normal Grunt weapon, but in different ways and emphasis. The BR steps up in cartridge power, while the DMR steps up in accuracy (but not necessarily in cartridge power!). For this reason a BR is just like a normal AEG in airsoft, while a DMR will have the best accuracy upgrades available while staying barely full-auto (and thus field) legal.