Zed is an assassin, and in most drawn out fights, he doesn't do too well (Or at least, he isn't supposed to).
Riven is an AD Caster, and rarely (if ever) builds attack speed, instead relying on her abilities and getting CDR.
Jax is a melee carry, just not a melee ADC (Which are terms you are confusing).
Xin Zhao can 100% be built as a melee ADC, but that is not how he is normally played, hence not being classified as a melee ADC.
The people you are talking about are Melee ADCs. which is very distinct class apart from everything else in the game. Melee carries are not in any way the same thing, except for both dealing high damage end game. Anything that is melee range and is your team's damage source end game is a melee carry, but that includes assassins and some AD bruisers. Melee ADCs are much more specifically melee versions of an ADC (kinda makes sense). They exclusively use AD, AS, and crit for damage (except for off builds). They are usually quite squishy and have built in mechanics to prevent their own death (Yi's Alpha strike and meditate, Fiora's parry and ult, Tryndamere's ult and heal, Yasuo's Shield and wind wall, GP's oranges), and all have some sort of free built in steroid, unlike true assassins or bruisers (Yi's double strike and ult, Fiora's free AD and burst of speed, Trynd's Crits and free AD, Yasuo's double crit, GP's Paaarrley and raise morale). The reason I said Xin could be a melee ADC but generally isn't grouped like that is that while he does have potent steroids, he doesn't have any innate ability to protect himself. Once he goes in, he stays in, and that makes him really unsafe to use as a melee ADC, since he would get blown up if built glass cannon.