I first heard the term, battle rifle, and assult rifle back in the early 80s in a book written by Chuck Taylor. I don't remember the name of the book, but I still have it somewhere. Basically, he took all the common battle rifles and evaluated them based on his own criteria. He also had a similar book on sub guns and fighting type shotguns. He detailed the drills he fired etc. As I remember it, it was a pretty good book. I don't remember all the rifles he evaluated but I remember there was the HK 91, FN FAL, M14, M1, whatever the Swiss have or had (Sig ? ).
Basically a battle rifle is a rifle that fires a full sized rifle cartridge. It may be select fire or not. I believe it also needs to be a full sized rifle and not a carbine to meet his definition.
An assult rifle fires an intermediate sized cartridge (less than a full bore rifle and bigger than a handgun). According to his definition, an assult rifle must be capable of select fire.
A submachine gun fires a pistol cartridge and obviously must be full auto or select fire.
There were other criteria in these definitions, but these are the main points. Personally, I like having words that have specific meanings. I see nothing wrong with classifying things into various groups. It makes communication easier when there are specific terms to define objects and people know these specific defnitions.
Oh, and back to the original question: An AR15 is not an assult rifle according to Chuck Taylor's definition since it is not select fire. It is simply a rifle or carbine. If you had an M16 in 6.8, it would still be an assult rifle in my opinion since it is not a full bore rifle cartridge.