I believe LoI or PoR may have been the first time it was officially called the Vampire Killer, but of course the name comes from the overseas release of the MSX2 Akumajo Dracula, making it seem like that was always the official name.
I have always understood the different forms to be shape-shifting upgrades that reflect the power of its user. It could be argued that every whip used in Simon's Quest is just a different form of the magic-infused Vampire Killer. In ordinary circumstances, it seems to remain in the form of a leather bullwhip, and as it gathers power, it will shift according to the will of its user. In Haunted Castle, it even morphs into a sword, which just seems weird to most CV fans. LoI definitely provides the most in-depth history of how the Whip of Alchemy became the Vampire Killer, but PoI goes into greater detail about how it's connected to the wielder's health and capability.