If I might compress and summarize the points in those two posts,
Brendan makes the case that the difference between old school, new school, and indie RPGs are the sorts of procedures the game has stated in the rules. OD&D and B/X have very low-level, granular procedures like determining if a trap is found, or whether an attack hits, but the games do not provide any procedures for actually playing the game. Actions like stocking up on supplies, gathering rumors, and going on adventures are implicit procedures that are passed from gamer to gamer by personal interaction. You might call them cultural procedures.
The new indie RPG scene uses procedures too, but at a higher level. The procedures tell you have to play the game with rules like "Roll dice or say 'Yes'." As Brendan describes, this makes the game easier to pick up just from reading it and you don't need the sort of personal interaction that is necessary to pick up what D&D is about quickly. But the downside is it limits the sorts of games you can play while still being true to the rules.
So far so good.
Arnold then picks up from there and tries to describe the difference between Proceduralsim versus what he calls "consensus" rule making. But personally I don't think this is a useful distinction. It's all procedures. The only question is whether they're in the book or if the group has to make them up on the spot. We call these house rules, and that's hardly a new concept.
So what's this post about?
The more interesting question to me, when comparing old school, new school, and indie games is when we stop needing procedures entirely and can rely on inductive logic.
Consider that you're playing an elven thief named Mariel. Here are four things you might have Mariel do-
1. Pick up the rock.
2. Throw rock at yonder goblin with intent to harm.
3. Hide the rock under a blanket so it cannot be found.
4. Convince the goblin guards the rock is a valuable bribe.
What's your chance of success of doing any of these things?
I would say that the primary difference between old school and indie games is when procedures take over from inductive reason.
Can Mariel pick up the rock?
In old school games this is always answered logically. The rock has a weight and the PC has a Strength score. Absent the use of levers and fulcrums, that's all you need to answer the question Yes or No. If the answer isn't obvious however you fall back to the procedure - make a Strength check.
Indie Games tend to rely on logic less and procedures more. Maybe a Test of Strength is always required (unless the answer is Yes).
Can Mariel hit the Goblin with the rock?
Pretty much all games rely on a procedure for combat. The simple reason is that you cannot inductively reason whether you win or lose, because the necessary information (like how well you swing your sword, or if you leave yourself open to a riposte) is not something the players can perceive and argue - the dice reveal this information.
Indie games aren't so different really, but they do tend to use different procedures than old school games. Old school games are more likely to allow the players to use their "player skill" to tip the odds in their favor, perhaps with clever strategy. New school games take this even further, giving more fine grain control over the tactics of combat. Indie games, generally speaking, go the other way and take a more hand-waivey approach to combat. It might be as simple as a single roll of the dice to decide a whole fight.
Looking at the above you might say "Well, it's all procedure then - no real difference". But that isn't so. Old school games still allow for more "logic" then the new school or indie games. A good example would be the improvised use of a non-combat spell or magic item in a combat situation. What would happen if you tricked a dragon into swallowing Daern's Instant Fortress and then yelling the command word? This situation isn't anticipated by the "effects" blurb in the item description so D&D IV rules lawyers might litigate this all to hell, and an indie game that resolves combat by flipping coins wouldn't even allow you try. But in AD&D? Kablooey, I'm thinking.
Can Mariel hide the rock under a blanket?
This is the question where old school and new school games diverge. In an old school game the DM and player would discuss how big the rock is, whether bumps would be obvious, etc. A new school game would probably just roll a Hide or Disguise check.
Will you let us go if we give you this very valuable rock?
Ah, role-playing. Or as D&D IV called it, Social Challenge. DC 28. And that's all I have to say on logic vs. procedure in this category.
So that's my input about procedure vs inductive logic. Where you draw the line between using one or the other is the key difference between indie and old school.
Actually, my conclusion was a lie. It's all procedure, from start to finish. Even OD&D is 100% procedure. And do you want to know what the old school procedure is?
Use your brain. Look at what your character is (stats, race, class, level) and what tools they have - and figure it out. And if you can't figure it out, roll some dice. Here's some dice-rolling practices we've developed to resolve the more common can't-figure-it-out situations.