D&D 3.5e isn't historically representative.
If you look at the picture of the swords in the Player's Handbook, you will find that the longsword is actually a picture of a bastard sword, the bastard sword is actually a picture of a longsword and the greatsword is actually a zweihander. There were no "greatswords" per se, it was just a generalization for D&D to classify big long heavy sword that requires two hands.
The bastard sword, or hand and a half sword, had a standard grip for one hand, but also tapered at the bottom with a hefty pommel that could be gripped by the second hand. If you look, once again at the Player's Handbook, that is the photo used by the longsword.
The longsword, historically, was not a one handed blade. The handle grip was designed for two hands. It was a lot lighter and more mobile than the two-handed sword.
When it comes to one handed swords, once again it is a generalization. The Anglo-Saxon Seax wasn't primarily a stabbing (piercing weapon). It was an edged weapon (slashing) and it has been noted as such due to the lack of handguard. Handguards, not only helped protect the fingers from another attack, but also prevented your hand from sliding up the blade when stabbing.
Look at this "short sword" (seax) from Owen Bush. There is very little to stop your hand from sliding off the grip if a stab is made. That would make stabbing with it less likely. Could you stab? Sure, why not. Is it optimal? No.
Look at this "short sword" (gladius) from Cult of Athena. There is a very pronounced handguard to prevent your hand from sliding forward. Also look at the very prominent piercing tip. That would make slicing with it less likely. Could you slice? Sure, why not. Is it optimal? No.
Reading the short swords description in the Player's Handbook will reveal that it is often used as an off-hand weapon. Due to the piercing nature and the european influence of D&D, the short sword in the player's handbook is most likely modeled after a main gauche.
Look at this main gauche from True Swords. It is a little longer than the typical dagger, with a very very deliberate pointed tip. Main gauches, in historical fencing, were designed to parry and stab in conjunction with a rapier, foil, epee, etc.
The wakizashi, in Oriental Adventures, is listed as having the same characteristics as a masterwork short sword. However, wakizashis were made along with and with similar techniques as the katana. One doesn't need to go far to see the cutting tests with the katana.
Look at this wakizashi from Casiberia. The wakizashi has the slight bend to the blade, emphasizing a cutting edge, rather than a piercing tip. Can you pierce with a wakizashi? Of course. Will it slice better? Absolutely.
Use Houserules for alternatives.
Wanting a thrusting short sword? Style it as a gladius. Want a slashing shortsword? Style it as a seax. Your culture and region of play in-game will determine what weapons are available, in use, and how they are used.
No, glaives aren't swords, and halberds aren't axes. Putting a blade on the end of a pole makes it fundamentally a different weapon. After all, a spear is not a dagger!
That said, if you're the DM (and you're not playing in Adventurers League), you can make substitutions in published adventures' treasure as you see fit. If you want to have the party find a Dragon Slaying Glaive instead of a Dragon Slaying Sword, do it. That sounds pretty cool.
Best Answer
Vorpal has a particular meaning in D&D
The term "vorpal" was imported into the game from Lewis Carroll's poem Jabberwocky.
The original authors of D&D borrowed this term and put it into the game. It's meaning was first spelled out in the Greyhawk Supplement (TSR, 1974, OD&D, p. 47). A vorpal blade was related to another magical sword, the sword of sharpness. Instead of just doing damage, they both lopped off the limbs or heads of whomever or whatever they hit, if the "to hit" score was exceeded by a certain amount on the d20 roll. (You can see in this magical feature a precursor to some later critical hit style conventions in the Blackmoor supplement).
Over the various editions of this game, some of the above has been revised or changed, but the core element remains the same: the vorpal blade / sword sometimes cuts off the target's head.
The vorpal ax is a viable adaptation, since an ax does slashing damage.
In D&D 5e, the vorpal sword is ...
After examining the Vorpal Sword (DMG, pg. 209)
There is no reason one could not adapt that to any weapon that does slashing damage. For example, the battle ax, great ax, or scimitar would fit perfectly.
About that deadly fork
The closest weapon to a fork I can find is:
A magical fork would be expected to do piercing damage; thus, a vorpal fork makes no sense in the context of that magical ability. If you want to make a legendary fork, I'd suggest using a different adjective to describe it rather than vorpal, since that term has a discrete in-game meaning.
What you describe is a different enchantment, however.
Murderous Fork, Fork of Lethality, Forking Assassin, Ur Forked or even The Last Utensil are suggested names; how to name it really should come from its creator. That's you. :)