I think the Scythe is a very appropriate name for the Slayer's weapon. Mainly because of the symbolism: Buffy, the Slayer, the Grim Reaper to all vampires and demon kind, the executioner of the forces of darkness--of course, the Slayer carries the Scythe. The name of her weapon is symbolic of her role. Besides the Grim Reaper, the Roman God Saturn was often depicted holding a scythe meant to symbolize his "cutting down and subverting all things." Sound familiar?
Buffy's weapon is her Scythe for its symbolism and how it denotes her role as the Slayer. And why would a weapon that predates history be called a weaponry term that was invented in the Middle Ages (e.g. lochebar axe, bardiche, etc, which by the way are pole axes, not melee weapons)?
A scythe is one of the most ancient weapons. A weapon that began as an agricultural tool, but then turned sinister. Again, fitting symbolism for a girl chosen to hunt and slay vampires.
Terminology of ancient weapons becomes almost moot when the oldest tool/weapon in human history is an axe which is little more than a rock with a sharp edge. If objects are given names to denote meaning, then the Slayer's weapon being called the Scythe has far more meaning than any other name. At generic best, one could call it an axe, but there is no real world weapon that is exactly like the Slayer's scythe (which is more than just an axe), so why should we expect it to have an accurate descriptive real world name?
The Scythe then is named not for its descriptive form (it's an axe in form), but is named for its function: reaping demons. A scythe is a tool used for reaping and so the Slayer's weapon is the Scythe. Even in the text, it's acknowledged that the weapon resembles an axe (Willow researching the history of the axe), but then they discover this:
(paces) "M" plus glottal stop is represented by a picture that's commonly thought to symbolize a sickle or a scythe. It's in thousands of carvings. In Egypt, throughout the ancient world.
A sickle or a scythe were two ancient forms of weapons. As Giles goes on to say, "a scythe is a symbol of death." The weapon is named for its symbolism, not for its axe shape. Recall also how in this scene, Willow is researching specific axes. They're not looking for a type of axe like a battle axe or a tomahawk. They're looking for one specific axe with a specific name.
Maybe we should research the weapon itself. Like...look. Maybe it's the Axe of Dekeron, said to have been forged in hell itself. Lost since the Children's Crusade, where it was said to have killed— (sits back, makes a face) Oh. Children. I hope that's not it.
Well, I have reference to the Sword of Moskva, the, uh...Reaper of the Tigris—how are we supposed to narrow this down?
The Scythe then isn't what form of weapon it is. It's the name of the weapon, named for what it does of symbolic importance (reaping demons), and not what it looks like (an axe with a stake at one end) or as Angel puts it, "that real cool axe-thing."
In Chosen, Mutant Enemy/Joss said it's an "axe-thing" so we as fans aren't earning supercool bonus points for being smarter than them in knowing it's an axe. (Likewise the motion comics don't get a free pass for failing to correct the mispronunciation of a word as "Sith" when that mistake was brought to their attention during production. Way to fail, FOX. Not only negligent, but lazy.)
So the Slayer's weapon is an "axe-thing" that's called the Scythe for being a "symbol of death." It's not a mistake, but a name full of meaning. Appropriate name is appropriate.