Many people in the firearms community today use the neologism "Battle Rifle" to refer to rifles firing "full-power rifle cartridges". Most often, however, the term isn't used for rifles like the M1 Garand or Mosin-Nagant (instead being rightfully referred to as a "semi-automatic rifle" or "bolt-action rifle" respectively) but only to Sturmgewehr 44 derived rifle designs chambered in "full-power rifle cartridges" like the FN FAL or H&K G3. Below, I will attempt to explain why I think this is incorrect and why.
The "Assault rifle" was first developed in Nazi Germany. Engineers imagined a rifle that would have the firepower and portability of a submachine gun, yet still keep the power and longer range of a rifle. This was because it was noticed that most infantry fighting was taken place at closer ranges than previous wars and the soldiers with the most firepower usually were the winners. The resulting weapon was officially known as the Machinenpistole 43 (submachine gun of 1943). It was designated a submachine gun because Hitler did not approve of a new rifle. It is said that on a visit to the eastern front in 1944, Hitler found out about the new weapon and, impressed by it, redesignated it the Sturmgewehr 44, "Sturmgewehr" translating to "storm rifle" or "assault rifle". A successor to the Sturmgewehr 44, the Sturmgewehr 45 was planned but ultimately canceled because of Germany's defeat during World War II.
Interested in the new weapon the Germans had developed, the British began development of their own assault rifle after the war. The "EM-2" was intended to replace the Lee-Enfield bolt-action rifle and various submachine guns in the British military. A new round was also developed to go along with the EM-2, the .280 British. Many other countries took interest in the new assault rifle, most notably Fabrique Nationale of Belgium. FN had also developed an assault rifle. This rifle came to be known as the FN FAL, and it too was chambered in the .280 British.
However, the Americans were not impressed with the power of the new rifles. Instead, they pushed NATO to adopt their 7.62x51 cartridge. It is unsure why NATO adopted the 7.62x51, even when almost every other country had accepted the "assault rifle" concept and had evidence that smaller caliber rifle rounds were more controllable in automatic fire. There is a rumor that Winston Churchill compromised with the US Military, that if they accepted the FN FAL as the standardized rifle of NATO they would accept the 7.62x51 round as NATO standard. Eventually the Americans went against their bargain, adopting the M14 rifle instead, muddling up a bunch of rifle projects being adopted in Europe. Eventually though, the Americans would be proven wrong, as the 7.62x51 was too powerful for automatic fire. NATO would adopt the 5.56x45 round as the standard rifle cartridge. The British and Germans had already knew a shortened rifle cartridge (an "intermediate" cartridge if you will) produced managable recoil in automatic firearms. NATO was already considering the 7.92 Kurz and .280 British cartridge before the US forced them to accept the 7.62 NATO.
I believe the term "Battle rifle" came about as a way to distinguish traditional rifles (nowadays also called "hunting rifles") like the M1 Garand, M1903 Springfield, Lee-Enfield, Karabiner 98k, M14 and Remington 700 (to name just a few) from the new breed of automatic carbines with pistol grips based off the STG44's design like the M16, AK-47, FN FAL, H&K G3, FAMAS, SA-80 and HK416 (once again, to name just a few). In my opinion, however, if anything, "assault rifles" and "battle rifles" should just be called "automatic rifles". As i've said before, "assault rifle" was a propagandistic term invented by Hitler, let alone there is no worldwide accepted definition of what characterizes an "assault rifle". Bottom line: an assault rifle is more or less just a layout/design in the same way newer rifles are in the "bullpup" layout.