DI is completely independent from direction; there is no such thing as "horizontal" or "vertical" DI. I think there's a lot of common misunderstanding about what a lot of this actually means and how game mechanics work so I'll explain as best I can within the mechanics of all four official smash games.
In smash games other than Smash 64, you can DI. This is just holding a direction on the frame you exit hitlag (that pause state that occurs as you are being hit), and what it does is bend the angle of the launch by about 20 degrees toward the direction you are holding (I think the exact difference it makes varies from game to game, but it's always in the neighborhood of 20 degrees). Obviously that doesn't mean anything if you are holding either directly toward or away from the launch so, for proper DI, you should hold a direction perpendicular to the launch. Some people suggested this was different in smash 4 than in other smash games, but to me, it felt exactly the same.
SDI is the ability to teleport during hitlag and is present in every smash game except apparently smash 4 but including smash 64. You just hit the stick in a direction and you teleport a small distance in that direction during hitlag. Notably, you don't have to reset the control stick to neutral to input another SDI so you can just wiggle the stick between the direction you want to go and one of the near diagonals so rapidly SDI even with low tech skill (the c-stick can be used for SDI, but it requires neutral resets so it's only useful for high tech skill double stick SDI). The teleport distance is mostly too small to make a significant survival difference so the main utility is escaping multi-hit moves; you probably notice strong Brawl players never get hit by the entirety of Pikachu's down smash, and SDI is the reason. However, with enough reactions and tech skill, it can really be used to neuter any multi-hit moves; top players in Melee use it to avoid the second hit of Fox's deadly up aerial with a terrifying success rate! Mostly this being gone from smash 4 means that multi-hit moves are a lot more viable when strong players are involved; in Brawl especially, so many of them were just terrible both due to poor parameterization in the first place (so many just didn't work normally) and due to SDI making them very practical to consistently escape and often even punish the hapless attacker.
aSDI is related to SDI closely. I don't know if it's in smash 64, but it's definitely in Melee and Brawl. You just get an extra and smaller "free" SDI on the frame hitlag ends in the direction you're holding on the control stick. Obviously this conflicts with DI, but it's just extra help escaping those multi-hit moves.
Air control is a mechanic present in every smash game that has nothing whatsoever to do with DI that is often confused with it. When in the air in most states (not hitstun, not executing most special moves), you can direct your character to the left or right by holding the control stick. This actually has three associated parameters to give each character unique physics. You have a character specific top speed, acceleration per frame, and a deceleration boost per frame. The first two are pretty straightforward; your character accelerates by so much per frame capped out at some max speed. The very high aerial mobility characters like Jigglypuff and Wario have very high values for both, but some characters mix and match (for instance, Yoshi traditionally has one of the best top speeds but a mediocre acceleration). The deceleration boost is a factor that causes most characters to lose speed faster than gaining it. That is, if you're already moving in one direction and then hit the stick in the other direction, you'll slow down to zero faster than the time it would take you to speed up from zero to max speed. This is a pretty small but real factor for most characters, but for a few, it's very large (Mr. Game & Watch in Brawl notably has a very large deceleration boost which is the main source of his excellent air control). As far as I can tell, typical values of this are pretty steady across the series though it's not always intuitive; a lot of people think of Kirby and Jigglypuff as similar characters, but while Jigglypuff always has excellent air control, Kirby's has always been quite poor which forces Kirby to rely on the instantaneous movement bonuses of mid-air jumps to move around effectively. No, from what I can tell from the demo, this is not a bit lower than it was on average in Brawl, but neither Jigglypuff nor Wario were in the demo who as a whole really were just so much better than every other character in this regard so who can say?
There's also fall speed which is also unrelated to everything else. Gravity constantly affects every character though in smash 4 it doesn't seem to while you are in hitstun (it does in every other game). You have a character specific acceleration due to gravity and a character specific terminal fall speed. Inputting a fast fall by pressing down at any point you are not actively rising from a jump and are in mid-air will cause you to begin to fastfall which adds some character specific value to your acceleration and increases your terminal velocity (again, character specific how much). Reports are very mixed on how these parameters vary between Brawl and Smash 4. It kinda seemed to me like fall speeds were a little higher on average and fast fall speeds were a lot higher on average, but it's such a character specific thing (could easily be one way with one character and exactly the opposite with another) and so impossible to measure in a demo that no one can make any good claims with confidence on this matter.
I hope that clears things up for everyone on how hitting the stick in directions while in the air or while being hit affects how you move in smash. Not only is understanding that very important to playing well, but it's also very important to communicating about the game.
