You're partially correct. As a rule, the US military does not use either "assault rifle" or "battle rifle" in weapon nomenclature, just "rifle".The military doesn't really use the term assault weapon/rifle. They have used Battle Rifles...
There are two exceptions to the rule: The Mk14 EBR (Enhanced Battle Rifle), and the Mk16/Mk17 SCAR (Special Operations Forces Combat Assault Rifle).
"Battle rifle" is an academic term used primarily by authors to distinguish select-fire, magazine-fed rifles that fire full-power ammo from those that fire intermediate-power ammo.