So there are all these fucking YouTube videos, right? Katana vs. European longsword. Some of these videos point at their own vapidness, but that doesn't matter 'cause they all miss the crucial fucking thing. One edge? Two? Straight, curved, or cheat both into the blade design? None of those things matter. Forget about them and check this shit out:
Perfect image -- and not a terribly uncommon type of shot in samurai cinema. Imagine this: If you were to start in the posture of the guy on the right, you could terminate your strike so that it resolves in the left posture. This is cool because of the traits that come from the overhead vertical strike; you strike very powerfully along the centre of the target, which has both offensive and defensive benefits. The stance on the left represents those defensive benefits, as you physically occupy the theoretical vertical line down the middle of your opponent -- the centre line.
Take that vertical line. If your sword reaches it before your opponent's does, then you're more likely to maintain control of that centre line, presenting a threat to your opponent while preventing them from crossing. That logic can be extended to each other line. An attack can only ever resolve along one line (although it can change line at points), but it can close multiple lines defensively depending upon its point of resolution. Now we come to the actual, key difference.
Note how this (gorgeous) longsword has hand guard traits like both a conventional longsword and a katana. The rounded element of this hand guard protects the hands as long as the sword is pointed forward; the cross element protect the hand by collecting an opponent's attack along the plane of the edge. Alone, the cross doesn't work well when held at vertical alignment, as it has no horizontal defenses against hand sniping (which is a huge deal); a purely horizontal alignment is better, but can be exploited vertically. A diagonal alignment, on the other hand, protects the hands extremely well. It's a very sensible choice if your fighting system already favours diagonal cuts -- but why would a system favour those?
If you look at the diagrams from before, you'll note that diagonal cuts will cross two axis; before any deliberate decision is taken into account, a diagonal motion of the sword is more likely to intercept an opponent's strike in some way. The cross guard is used in styles and regions more fond of the line denial generated by diagonal strikes and alignments; circular guards (or close enough equivalents, like basket hilts) prefer to occupy a line first and foremost. Perhaps the only downside to using a vertical alignment in the centre is the relative weakness of a sword's flat when defending against attacks, although a sword with a circular guard type will at least prevent immediate injury (as long as the sword is between you and them, and point in front of you).
It's worth noting that while the centre line is a present theoretical construct in European swordplay, it was seldom physically used due to the risk posed to one's hands, at least with a cross-guarded hilt design. Contrarily, Japanese swordplay uses literal centre line cuts on a very regular basis. Centre line techniques are so pervasive that they're generally among the first two cuts taught in Japanese systems, in fact, as well as being clear symbols of Japanese genres in some forms of fiction.
Blah blah blah curved blades blah tapered point blah blah folded blah fuck all that noise. This is the real shit right here. Before any of that claptrap even begins to matter, you have to control the space between yourself and your opponent while minimising risk. Controlling space and initiative are key; minor injuring traits of a weapon far less so.
So, basically, it's all in the technique and all this nattering about shape and steel quality obfuscates that, when all that really does is subtly change said technique.
Technique and systemic framework. The traits of an option usually represent deliberate, technical preferences within a region, culture, or style -- as you point out. There are exceptions, though, where the design really can be crucial. Blade length is probably the best example; when the blade length is roughly half your height or more in length, variations on one central parry can defend against any attack you have direct facing towards (see Left Samurai from before).
Shorter blades are at more risk of becoming entangled in committed parries, unless a shield is taking the bulk of that duty. On the other hand, longer weapons are more easily manipulated by opponents when struck or entangled near to the tip. All this is to say that sword design has elements of both technical requirement and technical preference. Many swords are essentially interchangeable in the hands of a good fencer, but you get substantial oddities fairly often, too.
It's difficult for me to explain with much depth. I'm no horseman.
In terms of raw sword design, though, the only common factor appears to be a taste for length. While the Roman spatha will end up being a very standard kind of infantry sword, it began life as a cavalry variant of the gladius; prior to the katana, Japanese horsemen used tachi, which are curved closer to the hilt of the sword than the tip (which is the inverse of katana design). Some European horsemen used rapiers, of all things, and some cultures had warriors who rode into battle with two-handed weapons.
This is a long-winded way of saying I'm not qualified on this matter as I am on generalist or infantry variants. I could feed you a bunch of semi-educated opinions, but that's about it.
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not disappointed
What about cavalry designs, though? That's an interesting area.