No, they are not the original names. -ra,-ga,-ja have been the standard in Japanese since FF1, the English translation had to use numbers because of limitations and (later on) simply poor translation work (see Aerith).

I don't understand how I,II,III is more intuitive in this game where spells have completely different purposes. Fire deals damage, Fire II does not deal more damage and is AoE (so the II there isn't helping anyone understand when this specific spell should be used), Fire III deals more damage but it's not supposed to be cast all the time (again, how is III helping here?). Numbers are only confusing and this applies to all magic spells, even cures. It made sense in past FF games where Thundara replaced Thunder and Thundaga was even better but here each spell is used even at 50 (not counting stuff like Freeze and Blizzard II).