Point being, there are several competing systems for classifying swords and each is just as valid as the next.
There's the hitch; that whole, "each is just as valid as the next." If someone's classification system is to call swords over 4' shortswords and those under 4' longswords and those exactly 4' purpleswords, I would not consider that to be "just as valid" as a system where a sword with 5" of grip or less and a blade no more than 2' long is a Shortsword, a sword with 5" of grip or less and a blade longer than 2' but no more than 4' is an Arming Sword, a sword with more than 5" and a blade that will neither strike the ground when rotated down from a proper stance nor impede the grip when grabbing an opponent at arm's length is a Longsword, a sword in between an Arming Sword and a Longsword is a Bastard Sword, a sword at the largest blade length that still qualifies as a Longsword is a War Sword, and a sword with a blade length long enough that it interferes with swings, parries, and other standard sword techniques and must be used more like a polearm than an actual sword (despite still having the profile of a sword) is a Greatsword.
The names of the weapons are more for the benefit of us as players. Your character might just call his weapon a "sword", but in order to adjudicate a game system, we need a good system of classifying weapons. The system that Pathfinder uses, which started mainly with the original D&D, isn't exactly "bad", but I'd say it isn't exactly "good" either.
But, basically, you have 4 major categories of a "hand-held sword-like bladed weapon"; Daggers, Arming Swords, Longswords, Greatswords. Daggers are small and easy to use. Arming Swords have a one-handed grip and are intended to be used as a backup weapon with a shield (for when you lose your primary weapons, your lance and mace/flail). Longswords are named for their longer grip, able to easily accommodate two hands and would be used without a shield but still can be one-handed for free-hand maneuvers. Lastly, Greatswords are so big that they only have the general profile of a sword; they can't be used with most sword techniques. Then you have the hybrid weapons; a shortsword is between a dagger and an arming sword. A Bastard Sword is a longsword that is very close to an arming sword. A War Sword is a longsword that is very close to a Greatsword. And it doesn't help that most RPG systems have a big case of anachronism stew with weaponry spanning several ages, all the way from bronze age weapons to Renaissance and everything in between.