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.
As written, it does not affect the versatile damage.
The description of versatile says (PHB p.147):
Versatile. This weapon can be used with one or two
hands. A damage value in parentheses appears with the
property—the damage when the weapon is used with
two hands to make a melee attack.
The spell does exactly and only what it says "the weapon's damage die becomes a d8". So a quarterstaff does 1d8 damage 1 handed and remains a versatile weapon doing ... 1d8 damage when wielded 2 handed.
Is it "feasible" to say it increases the damage? No. Is it "reasonable" to do so? Why not, its not going to break anything and it makes enough sense that I'm not going to come around to your house and tell you "You're playing it wrong!"
Best Answer
It's a scimitar.
Don't let the name fool you - what D&D calls a scimitar is more properly a machete, parang or kukhri. It's a light weapon that can be used with finesse.