>TUGOWU
A fine place, but somewhat troublesome to find all one wants. However, a lot of history didn't get to take place there (GS, GFAQs, alt.TUGOWU, Vet Board, GoW3forums, etc.). GoW1 and GoWII (to an extent, CoO) should be fine (GoW:A's stock increased majorly upon my return, but it still had SBK before that), but GoWIII and GoS? There won't be a lot there (and the former is what you're working with).
>some guy's PAIN
Don't know him anymore (though apparently I once did). Can't speak to his tactics (without going over things again), if he followed the rules (so many *upon review* violate the rules during Box 6 in particular), if it was PAIN or PAIN+ (people often don't name things correctly), etc. At a glimpse, it is PAIN+ and I only did a very light evaluation of these (very) early tactics.
>Shin's NUR+
I often advise caution when emulating him owed to his habit of relying on his skill over good tactics. He got better towards the end, but usually I still suggest Taz instead.
>put away for now
You're mostly focused on taking advantage of Power % (as you are on H, you are still at 100 *not 75*) and upgraded modifiers for your default weapon. You won't really be getting a ton of 't,t,t' (or 't,t,s') from either (as they don't have it). They can't use any counters (besides Return *well, Shin can*), either. You need to take advantage of the options you have (including the lazy early face-tanking option that is Rage of Sparta *when out of better resources and wanting to rush for victory while allowed to do so*). Shin will hardly make use of MP (if he hasn't *as usual* opted to rule out its use beyond the rules of the run) let alone spam it effectively to make things easier (so many chests to work).
With HCB, I imagine both launch (Shin probably using the Boots *you should probably switch to the whips for their t1*) Satyrs for air OS though keeping the one from S2 barely alive for S3 (so they can quickly kill it there to prevent x2 satyrs backing the mini-boss for long). I don't see Shin dumping MP on HCB (and the goat) a bit during W2 to speed it along and again in W3 (with RoS to seal the deal *to be left with just fauns*). You have the ability to power through things to a greater degree. While it is true Nemean Roar scales with Power % (so it should be a solid choice here *if you upgrade it to measure up against AoS from BoE*), the panic button AoE type is good enough for hitting everyone nearby in this case. You should have plenty of orbs to spare after MAX BoE, so it is entirely your call where to put the rest. There will be chests after the fight, so go wild (especially on H as orbs are just nerfed to 75% rather than 25% *and enemy strikes are at 250% instead of 500% while they attack about half as often in your case, so tanking with your super state won't lead to as much damage needing to be soaked as a concern*).
>make a dictionary
A 'legend' of sorts? I'll probably never really do that, but I can answer some things ahead of time (and as they are brought up).
>what you recall
Mostly correct. Not sure why CR is there (other than calling Nemesis Rage *Nemean Roar also shares NR* projectile CR-lite).
>second hit
Second input, rather. There will be cases like s4 in the BoE light string. That move hits twice on a single target and is the fourth part of that dialed up combo (s5 is the same, but with vertical instead of horizontal strikes).
CG is Combat Grapple which is a generic for all L1+o moves in GoWIII. For BoE, the move's proper name is Hyperion Ram (but you won't see much talk of HR). The generic grab was named officially in this entry. The hug became Orion's Snare (OS). GoW1 had already named the ground-to-air grab Orion's Harpoon (OH). As of GoWII, there was the option to hold the input for another version of 'Slam' type OH on most foes. This was called alt.OH, but it isn't really much of a factor in GoWIII (only exists on harpies who are not a big part of the game *not like collisions are all that they used to be, either*).
Boundary Abuse tends to refer to going to a spot foes are limited (or even unable) at. Be that they unexist while you are there, die instantly, AI breaks, they 'leash' (slowly make their way back to where they belong), physically CANNOT go there, have limited attacks, just like to taunt at you or whatever. Sweet spots are part of this (areas where enemies aren't quite able to fight back). Taunt provocation is more for VH, but you can still get use out of fighting from places the enemy cannot follow (moving away from this, just changing the venue can offer other advantages like opening up use of a hazard, restricting movement for them, giving more mobility to us, etc.).
I-frames are invincible (mostly) parts of an animation. 'Tricking' is exploitation of them (especially used to refer to non-evade instances). Just a damage negation matter. Learning the various names of attacks relevant to the run might be good. As you are not fussed with Items, I'll put them aside for now. Likewise, Relics. With BoE, s6 is Valor (of Hercules *this entry dubs the long-way to enders through the light string Olympic Fury, but that is mostly trivia as it is not able to help us name the parts along the way*). Spirit (of Hercules) is the 'Super Plume' t3. Plume (of Prometheus) is t0 (both t0 and t3 have a fake second hit that can register *in some entries there isn't one and in others the extra hit is real*). T is (Olympic) Ascension. A later entry dubs t1 as Bravery (of Hercules).
CoC (Cyclone of Chaos) is pretty whatever (though the improved ground version is cute for the lowest level foes) as the dps just isn't high enough to justify the risky dedicated animation. Hyperion Fury is a poor man's RotT BoA s3 playing at being GoW1 BoC's Might (of Hercules). Don't believe it offers the option to go right into 'Fury' (ender without the build-up slaps). Needed that and to be stronger (to go along with the better coverage). Certainly can't get fancy and work the build-up into a heavy input (for another path to Spirit unlike with MoH). Athena's Wrath is crap. Tartarus Rage is okay, but there isn't much of a good reason to use it (though the improved version has a small time to shine if able to get the air version stuck on a big target or mob beneath you *tons of bonus hits and probably not being hit back*). Doing an air input just before landing can also give an on-demand explosion (the ender), which is something. These are still dedicated (not able to cancel normally as with blocking, evading, etc. *though sometimes GoW allows things like rage, spells, etc. to do the trick*) moves, so...
Army of Sparta (AoS) is your spell with BoE. It will mostly be for clearing out mobs when you don't feel like fussing though it can also be used to lazily speed up some boss bits (doing damage while not at risk). Panic button implying the use of long i-frames to get a break in the action. Can't especially count on it to set-up RO (ring-out *sending things off cliffs* or PP glitch, though possible). If feeling less lazy, you can often weaponize the most basic foes using their grab option for running around with them called Battering Ram (BR) by the devs. To cover it (now), PP glitch is Partial Petrification (stoning). Normally enemies need to be fully petrified to become statues with their own HP value (different from their total, not altered by how much you hurt them up to that point or previous sessions with the statue even). You beat on them to shatter them (or manage it in the air and they break on-landing). GoWIII has no friendly fire on statues (where it used to be that enemies lose their normal protection from each other's strikes in that state *making it easy to weaponize them against each other*). The balancing of the mechanic is not as good (weaker than before), but this bug works for them (and against us).
In our case, it isn't so great to have a little build-up while in the air meaning you will die instantly upon hitting the ground unless doing certain things (like gliding *which works even on a full statue in this entry*). Could just blow off the concern with RoS activation (the state is immune even to build-up aside from breaking out of full petrification or some grabs we can struggle during). Enemies? If they are in the air (not in a flight state, mind you *I don't recall harpies with build-up doing bomber and exploding on impact*), get a little on them (probably not going to come from you *would need upgraded CoH and a lot of MP to spare*) and have a gorgon variant at the ready to douse them. Just a bit while they are off the ground in most situations (sliding knockback into wall splat being one exception) will lead to their instant death upon hitting the floor. Even a cyclops can get this treatment (anything that can petrify so long as it is found off the ground). Catching it in the air isn't going to happen in this entry, so you have to cause that (which is a trick unto itself). Usually this is just about using hit properties like knockback, bounce or launch on foes open to them and having a little of the stuff touch them before they land.
NOTE: Don't fret over L1+x attacks. They are terrible. The only useful case is one with a flailing from the whips when you use a 'swap' trick to have Cestus do that instead (against Onyx shields for a quick triple hit *enough to shatter them in one go*). An advanced maneuver you can attempt to learn (for laughs) another time as with CG poking (using Cestus), Cestus glitch (applying its properties to things that are not it), Cestus 'Homerun' (glitched CG), etc.
I'm sure aggro needs no explanation (nor even enviro). Collisions with respect to how the term will be used here is just in reference to when you make a foe collide with another foe in some meaningful fashion (though it can also be a thing with an object into an enemy or enemy into a wall). If you knockback, slam, toss, etc. one fellow into another. In GoWIII, there are three different cases for when this happens. Agent ONLY (rarest), Target ONLY (most common) and Agent/Target. If I remember right, there are full (10) and half (5) instances of all (except that agent/target only has half cases). Walls are just 1 (as I believe is also true of object instances like rapid pushing something into an enemy). Here, an "agent" is the thing acted upon by us with the "target" being what the former is made to collide with (collider and collidee respectively *if you will*). Often we'll be referring to those we can BR with our ammo, too (smart for higher runs to keep them alive until fully used up as a resource for damaging other enemies of slightly more worth *oftentimes*).
NOTE: Despite not really caring about Items/Relics, you should still probably learn a few names. We'll get to that before long. As before, I usually tackle things as I go (and as questions are asked *showing participation from the one being evaluated*).