Everyone has their own ideas & definitions, so I'll throw my hat into the ring here. Speaking as a long time LoL player I've seen the trends come and go. First off, while I'm not defending the meta lane assignment against all possibilities, it is something that was built up over time and has been reinforced as a core strategy.
See, originally, LoL didn't have these rigid lane assignments. You had 2 top, 1 mid, 2 bot. Or you had a jungler, sometimes, but not always. This was because Junglers didn't have special items to make their clears faster, they only had 1 smite at a time, they didn't have fancy smite effects.. all in all, they tended to be behind the curve in terms of gold. If you had a jungler, Mid & Top would be anywhere from 1-3 ahead of everyone, the 2 Bots would be about the same level and the Jungler would be somewhere in the middle. So you didn't always have a jungler.
However over time patterns began to emerge. Because bot lane shared XP (and often gold, since again there wasn't a set "Support" role), they were usually a bit behind the rest of the team. So the more you could isolate players the more XP they'd get, they'd be stronger and outscale their opponents. This is why a 1v2 top lane might win if their Champion scaled really well or if they had a kit that let them hold their own while outnumbered (Old Mordekaiser and Dodge Jax were my two favorite 1v2 to planers).
So eventually junglers became more and more common to give their tops the edge - or at least, avoid being behind because they were sharing XP. The jungler struggled in terms of items until late game, so Junglers were typically based on their ganking potential - Warwick's Ultimate made him one of the original Terror junglers once he hit 6, where Amumu might pose an even greater threat early on because of his long-range 1. Also, these Champions typically didn't need a ton of itemization; they had strong abilities (either their base damage was pretty high, or their abilities themselves had strong effects to help them jungle, such as CC for ganking or Healing to sustain in the jungle).
This mentality eventually made its way to bot lane, too. If you gave the ADC all of the minion kills (CS) to give them more gold, they'd be able to destroy a lane with 2 people who were sharing the gold. Early game, one big item is better than 2 small items.
Fast forward 5 years or so.. and now we have a unique set of Jungler & Support items that cater to their roles. Because of this the diversity of these roles have definitely increased. However, at its core, the Support Champion still does the same thing. This is how I personally define it:
The Support should help the ADC win the lane
Now there are a lot of different ways to win. Soraka wins the lane by giving her ADC more health, keeping them healthy and letting them win trades over and over until they push out the other person, or just kill them. Blitz wins the lane because his hook is a very strong ability, essentially letting him turn the fight into a 1v2 very briefly. He pulls the ADC away from their teammate and behind the minion line, which puts them in a very bad spot - especially if they already have the upper hand. He's also great at stopping them from retreating. He's the "I'll hold 'im, you punch!" support. However other Champions have started making an appearance; Morgana's long range snare lets her catch the enemy out of position, and Annie has a regular stun that can be targeted or AoE, in addition to high base damage.
Braum is the kind of Tanky support that is good at keeping at teammate alive (since his shield can block all projectiles, which can completely shut down someone like Jinx), whereas Leona is the kind of Tanky support that is good at starting a fight.
The common denominator between these Champions usually comes down to their base abilities. Generally, a Support needs to be able to do well with a limited amount of gold, especially if their lane doesn't get any early kills. Even though they have gold generation items these still won't compare to the CS gold from another lane. So maybe they have oodles of CC in their abilities, or they have shields & heals to keep their teammate alive. These all work, and they don't require having items to deal damage, they just use the "effects" of their abilities.
So what about "Kill Lanes" - where you take another DPSer or someone who deals damage in lane? Well, sure. If it helps them win the lane, those can help. Since "traditional" supports tend not to deal a lot of damage, a damage-based Support can sometimes tilt the lane by pushing both of the laners off the XP line and starve them. However these lanes will typically fall off if they don't get those early kills, and the lack of items will be a problem in the mid game until the 'support' has time to farm.