You sound like more of a "visual" learner than a "verbal" one, so I've drawn a picture for you.
So what we have here are two categories, "DI" and "SDI". As we see, the actual mechanic behind each is different: DI influences trajectory by the changing the "direction," while SDI influences trajectory by "warping." Both terms use "directional" to describe these mechanics, as we can see in the middle there: both DI and SDI have "DI" in the name.
This image is a bit more confusing but I'll walk you through it. The first two circles on the left are the same ones from the last image, but I've separated them from each other, and pulled "DI" as the larger category out to the left. I've also changed the mechanic name of DI from "direction" to "rotational vectors," which is the term Strong Bad used to describe it (and it seems true enough). So, "directional" collectively refers to the different mechanisms in both DI and SDI. This is separate form Smash 4's mechanic.
Now, on the right (purple) we see Smash 4's new mechanic, which uses additive vectors instead of rotational vectors. This is a new mechanic than the last two, but the question here is do we need a new name to describe it?
My point in this image is that, if you are going to argue that the new mechanic should be called "vectoring" so as to distinguish it from the old "DI", then that doesn't really make sense, since "vector" is a word that can describe both! We could have even used different terms long ago (called DI "VI") and then made a note about it instead.
Here's where I think we have ended up. As you can see, "DI" is used in both Melee's/Brawl's DI as well as SDI, and while there may be a separate term (on the right) to describe the functionality of Smash 4's mechanic (which is fine), I think, on the whole, we should refer to all the mechanics collectively as "DI", or if you think it best, we should come up with a new term.
"Vectoring," I think, is a silly, too-smart-sounding name that goes into the technical details of the game without capturing the overall picture.
Conda makes a good point:
While the counter to this may be that the confusion the two DIs would bring would outweigh any clarity that keeping the old term would preserve, using new, equally-confusing term wouldn't help either.
Since keeping "DI"
may cause this confusion, and since I think the Smash community can evolve with its terms, I really think we should call it "FI" for "flight influence," as that is the most neutral and all-encompassing term we have.