A "Battle Rifle" is generally a full-size semi-automatic or select-fire rifle that fires a full-power rifle cartridge, and often has a detachable box magazine, etc. Some full-power rifle cartridges used in battle rifles of the past include .30-06, 8x57JS, .308, 7.62x54R, 7.5x55 Swiss, 7x57, to the best of my knowledge. Examples include the M14/M1A, FN-FAL, SIG 542, SIG 510, Dragunov, HK G3, Japan type 64, AR-10, SR-25.
An "Assault Rifle" (non-legal definition) is generally a full-size or carbine-size select-fire rifle that shoots an intermediate-power rifle cartridge. Some examples of internediate cartridges used in assault rifles include: 5.56x45, 7.62x39, 5.45x39, . Examples of assault rifles include: M16/M4, AR70, AR70/90, Bofors AK5, AK-47/74, CZ58, Daewoo K2, Enfield L85A1, FAMAS, FARA 83, FN-CAL, FN-FNC, Galil, HK33E, HK G41, Imbel MD2, Japan Type 89, MP44, Sako M90, SIG 540, SIG 550, SR88A, Steyr AUG, Valmet M76/78.
That said, the only full-power rifle cartridge you listed was .308, so that's the one.
There have been numerous other threads discussing the difference between those calibers...
-z