Originally Posted by
Talraen
No. Dark Knights were not in FF3 until the DS release, 16 years after the original. In the original, they were "Demon Swordsman" or "Magic Swordsman" (the actual Japanese name being Makenshi, as opposed to the Japanese name for Dark Knights, Ankoku Kishi), though the most popular fan translation called them Mystic Knights, and they used level 1-3 white magic (which is why you can't really consider them Dark Knights). Their weapons are all katana, with the exception of the Moonring Blade and Ragnarok sword.
Cecil uses the same types of swords (curved and single-bladed; you can tell because of the slight extension at the top of the combat sword sprite) as FFIII Mystic Knights, which makes sense because FFIV is basically FFIII fanfic. However, they've never been given Japanese names like other katana in the series in any English translation, and Edge wields the swords with those typical names.
Pretty much all of this has been retconned into what you said, but I prefer to study the originals. :P