Sure, but consider this: in D&D you have on the one hand the shortsword, a light finesse weapon, analogous to the Roman gladius and other ancient swords, and on the other hand you have the versatile longsword. There's a gap between the two that represents the majority of sword evolution in the medieval period: the spatha and its descendants, the migration period, knightly, and arming swords. They were neither light (because you couldn't readily wield two of them) nor versatile (because they weren't made to be wielded with two hands). The rapier fills that gap in the D&D rules by losing the light property of its smaller cousin in exchange for a larger damage die, while it retains finesse. I'm inclined to treat all other swords in the one-handed arming sword family the same, because otherwise they would simply be the equivalent of longswords that forgo versatile for no other benefit.
I'd still count arming swords, viking-era swords etc as D&D longswords. You can still get a bit of extra force out of them by using two hands even if they aren't designed as two-handed weapons. In the same way that many bastard swords aren't designed to be used with one hand whilst on foot, but you technically could if you had to.
If a character tends to use their longsword one-handed, I'll probably assume its closer to an arming sword. If they generally use it two-handed, its probably closer to a historical longsword. I don't think that 5th ed has or needs enough granularity to worry about finer distinctions.
If needs be, think about an arming sword as versatile in that it can be, and often was, half-sworded using both hands.
Actually, I'd probably give it light myself, due to the emphasis on dual wielding pata or pairing it with other weapons. So basically a shortsword, although some of them could be quite long.
AFAIK the pata was more often paired with a small shield. Considering that it generally had a rather substantial blade attached to it, I'd probably treat it as a non-versatile longsword.
Depending on point of view, one could argue that the katana is a finesse weapon. It is very light and sharp. While I'm not exactly a katana fanboy (I prefer Western-style swords), I do have to give it credit for that. Caution should be exercised, though, as you're not far away from a versatile, finesse weapon, at that point.
Katana are generally rather heavy for their length actually, with the weight distributed towards the blade. While they look less deep in the blade than equivalent swords, they are generally thicker. This means that they are around the same weight as some bastard swords, but shorter. They are extremely sharp, but the style emphasises large sweeping cuts using the power of the shoulders rather than more finesse.
Given that katana are often equivalent length or shorter than scimitars, sabres, tulwars etc, you could use scimitar stats for them if you wanted finesse, with the paired katana and wakizashi treated as two scimitars.