The palatini were the second most prestigious "level" in the late Roman military hierarchy, ranking below the scholae, the emperor's mounted bodyguard following the disbandment of the Praetorian Guard. The word derives from palatium, quite simply meaning palace (after the Palatine Hill in Rome), and indicates their original status as the emperor's personal army. Later on they would become mixed in with the comitatenses, the regular field armies, but would retain a higher status than their peers. Palatina is simply another grammatical form of palatini (or rather palatinus), used for example in the auxilia palatina, the auxiliary arm of the palatini.

Paladin is commonly used to refer to a number of great knights or warriors serving Charlemagne, also known as the Twelve Peers. They appear in many French heroic sagas, but their actual historicity is dubious. As you probably suspected by now, the word does indeed share linguistic roots with palatini, deriving from Palatinus which referred to high-ranking officials serving the Roman Emperor in the imperial palace on the Palatine Hill. Ultimately it's a linguistic question rather than a historical one.