> of all of the battle rifles designed (The big three being the AR, AK and FAL/SCAR)
To be pedantic about terminology the Kalashnikov family of rifles are generally not Battle Rifles. Of your listed examples only the FN FAL and some models of the FN SCAR are considered Battle Rifles.
Battle Rifles as a term came out in the 80s to distinguish between autoloading rifles soldiers had fielded since WW2 that weren't assault rifles. Assault rifles are defined as a selective fire rifle using an intermediate cartridge with a detachable magazine.
An intermediate cartridge being one that is more powerful than a pistol but less so than a traditional rifle cartridge. Germany pioneered, or at least mass-adopted, this concept with the 7.92 Kurz [short] round in WW2. The USSR ran with this concept - through seeing the STG-44 (German 7.92 kurz assault rifle), capturing some German weapon designers and following the train of thought of the period. This ended up with the AK-47, then later variants on this pattern.
NATO countries generally also wanted to do this, see the .280 British - the original cartridge of the FAL (and the EM-2). With one big exception - the US Army Ordnance Corps pushed for adoption of a traditional cartridge, and NATO bowed (in part as the US agreed to adopt the FAL) and everyone ended up with the full powered 7.62 NATO round.
Despite pushing for it, the US soon found out that having a high powered cartridge (and the sort of weapon that goes with it) wasn't what they actually needed, and they chose the 5.56 NATO round as used in the original M16 rifle (AR-15 platform). The rest of NATO caught up soon enough, often with platforms based of the AR-18/AR-180. This left them in the 80's with a new type of rifle that we would call an Assault Rifle but they also needed to refer to the (more) modern rifles they'd been fielding since the 2nd World War.
This resulted in the term Battle Rifle being created to refer to a self-loading rifle using a full power cartridge with a slew of (optional) features like full auto (the SLR [UK's inch pattern FAL] was semi-auto only), pistol grips (the M14 had a traditional stock) and the like. The 7.62 NATO calibre versions of the FN SCAR are generally considered Battle Rifles, whereas the 5.56 NATO versions aren't.