The fighters you described are the assassin esque variant. Xin, Irelia, and Jax being three of them. They walk between the blurry line of melee carry with their defense steroids being balls of stats, massive tenacity reduction, and complete avoidance of auto attacks. There are a few more like them but those 3 stand out the most within their odd subset because of how strong they are when they're able to do what they do.
Melee carries aren't meant to be glass cannons, juggernauts should be a big reminder of this. Melee carries are characters who're able to output enough sustained damage to hold their and a few more people's weight in a teamfight with items. That's when League's character designs start to get thrown out of whack, especially in the fighter department. Quite a few of them, mostly earlier development, were designed to be melee carries who aren't as easily dealt with in fights like Master Yi(who at the time was significantly less reliable as he is today) and Tryndamere(whose design is a failure in my opinion, even post rework). Instead of Meditate or Undying Fury, they were given a moderate amount of CC, a good amount of stats, and a lot of single target damage that isn't on a long cooldown. Can't forget the mandatory self heal and long cooldown AOE damage ult.
Assassins are anti-carries. They go in, pick off their target, go out. Fighters that can reliably do an assassin's job border the line of semi squishy melee carry because they have little in the way of getting out, so their kits are loaded with "you can do more if you stay in the moment."