Poke someone a billion times with a feather and they will probably be dead. That's not the point, though. The divide in the real world and in the game world in prior editions is a reasonable likelihood to cause death. Poking someone with a feather isn't going to reasonably result in death, so it will be considered non-lethal damage. The same with a punch. That shifts when you look at master martial artists who know where and how to strike to kill with a punch. A normal person doesn't have a reasonable expectation of causing death if he punches someone.
Feathers run into the 'granularity problem' of hit point abstraction, but of course you can expect to be damaged/killed by enough feathers. Again, no such distinction as 'non-lethal' damage exists. As for punching, I beg to differ! A punch to the head is quite dangerous and MANY people have died from them. Usually the puncher claims lack of homicidal intent, but that is either ignorance or self-serving. I knew a kid in grade school that died from a punch. Ain't even that hard to accomplish. Beyond that, the 10th punch to the head? Yeah, tell me that isn't lethal damage.
It's like the difference between a penny and a $100 bill. Sure they are both money, but you can only reasonably expect to be able to buy things with the $100 bill. That penny isn't likely to be able to purchase things on its own.
It is still money. Anyway, what is this analogy accomplishing? The penny is an analogy for what? A sword? And the $100 bill is an analogy for a fireball? Both of them are serious weapons in a D&D game both can absolutely kill. Generally it is probably more likely in 'classic' D&D that the later would accomplish killing someone, all other things being equal, but that doesn't mean fighters don't expect their sword blows to kill.
If they didn't have that division, you couldn't knock anyone out. There must be both lethal and non-lethal damage types in order for you to both strike to kill and strike to knock out.
No, there need only be a provision for damage to knock people out instead of killing them. Your statement here is literally counterfactual. It bears no resemblance to the actual game it purports to describe.
This is from the 1e DMG.
"Striking To Subdue: This is effective against some monsters (and other creatures of humanoid size and type) OS indicated in the MONSTER MANUAL (under DRAGONS) or herein. Such attacks use the flat, butt, haft, pommel, or otherwise non-lethal parts of the weapons concerned but are otherwise the same as other attacks. Note that unless expressly stated otherwise, all subduing damage is 75% temporary, but 25% of such damage is actually damaging to the creature being subdued. This means that if 40 hit points of subduing damage has been inflicted upon an opponent, the creature has actually suffered 10 hit points of real damage. The above, of course, does not apply to player characters."
Then there was the two page combat section in the 1e DMG entitled, Non-lethal and Weaponless Combat Procedures.
It is a very wooly and almost never used thing, that doesn't even hold water when you start trying to use it. First of all it claims to describe some subset of monsters (why a subset?) which can be subdued. That tells me immediately there isn't a 'special kind of damage' but that there is a special kind of MONSTER! Note how PCs are explicitly excluded from this rule! So, while Gygax uses a phrase 'subduing damage' that IMHO doesn't (and we never played as if it did) suggest that the damage itself was of a different type, only that the intent was different.
So yes, lethal and non-lethal damage was in classic D&D against creatures other than dragons, and not as an optional rule(though all rules are technically optional).
And calling it a 'rule' is pretty dubious. I mean, technically maybe, but that whole system, which is mainly drawn from The Dragon #11 IIRC, is completely bonkers and should NEVER be used in play (it clearly wasn't playtested for even 5 minutes). It is basically throwaway filler, one of the few such things in 1e core books. Attacking to subdue clearly is a function of some sort of 'morale' (though oddly disjointed from the normal morale rules, great thing about Gary and rules...) because it works differently on different targets, etc. I just don't buy that there is anything besides 'real damage', even in AD&D. The very fact that it describes pulling blows and such, and the actual damage is 25% says it all to me. You can still kill with these attacks! Some monsters are just bound to surrender when they are used long before that happens.