There is no Hardtype, only an EasyType. "HardType" is basically the original game. When IV was originally released it was pretty challenging, so Square released an Easy version which had an altered sprite for the final boss and changed a few things around. The SNES version of IV (FFII for you oldbies) is similar in difficulty with Easy Type but is still based on the original IV just with a lot of content stripped out and censorship. Easy Type is Japanese exclusive and has never been ported to the West officially, though some sprite assets from that game were re-used for the Lunar Trials in FFIV Advance. The PSX port of IV is the original Japanese version of IV just with all the game content and censorship changes restored as well as the difficulty. So the PSX version is technically "Hard Type" though no such moniker actually exists, it's more of a fan nickname.