Partial cover and concealment are identical to cover and concealment.
It's a confusing terminology change that happened part-way through 4e's lifespan, not the addition of an extra level of cover and concealment. The modifier "partial" was added to 'normal' concealment --probably in a well-meaning effort to clarify things-- and both the Rules Compendium and the Online Compendium parenthetically note that it is "sometimes simply called 'concealment.'" (RC 222)
The Online Compendium's glossary entry on cover still hasn't appended the modifier "partial," nor does it note that this word might sometimes be added, but the Rules Compendium uses the phrase "partial cover" and adds that it is "sometimes simply called 'cover.'" (RC 220)
Keep an eye on "superior" and "total," too.
There is equal potential for confusion in the upper echelons of cover and concealment: better cover is called "superior cover," while excellent concealment is "total concealment." I see no particular reason these two similar effects should have different modifying terms, and am surprised that only one item (the Nightmare's Keen Senses from DR 393) is currently confused by this (according to a search of the Online Compendium at the time of this post).
A bit of history
"Story Game" has been used in many different ways, but at least in the context to Dungeon World, it has a definite lineage.
The term as associated use today, was first coined by Clinton R. Nixon (I believe around 2006-2007?) as a simple and catchy term for Narrativist games. This allowed a way to promote these types of games without having to deal with the baggage (social, terminology) of Forge Forums' GNS Theory (now "Big Model Theory" as developed by Ron Edwards).
It caught on with a subset of the Forge crowd and became the inspiration for the Story Games Forums where a sizeable chunk of that crowd ended up migrating to. As they kept producing games, or talking about games developed from the Forge/SG crowd, "story game" got applied to a lot of games, regardless of whether it was Narrativist focused or not. Basically the term ended up getting broadly applied for many things much like "indie" has been for the last several years.
How it gets used now
Well, "story-game" usually gets used to mean ANY one of the following:
A Narrativist focused game
A game that focuses on fictional elements over mechanics (which usually means inclusive of many Simulationist games, particularly if they are rules light)
A game developed by regular members of Story Games Forums or the Forge Forums
A game that is designed with a focus
A game that is rules light
A game that is innovative or different than whatever folks consider "Traditional"
A game that can be played in short form
Is Dungeon World a Story Game?
Well, there's a lot of potential definitions up there. If you ask most of the Story Games Forum crowd, they'd probably say yes, since it falls into the usual definitions they tend to use more often with it.
As you can see, though, there's a lot of options and no definite answer. Depending on what you mean by Story Game, maybe it is, maybe it isn't.
It's definitely focus designed, developed by the Forge/Storygame crowd and influence, it's different in some ways than traditional games (especially in the GM advice/hard rules for GMs, as well as the improv nature) but it's very traditional in the way it treats player/GM power divide and events.
Is it narrativist? I know Apocalypse World is, but that's because AW pushes hard moral decisions and character exploration, while I haven't had a chance to look close at Dungeon World's details to say.
Story Game vs. Storytelling Game
"Storytelling" is a term used nearly everywhere, and it, too, has a bunch of definitions. Overall, the problem is you're asking for hard definitions from terms people just kind of throw around and mean a lot of different things with.
On the other hand, "storytelling" doesn't have the same connotations as "Story Game" for the people who use the latter the most. Given how poorly they're both defined, neither do a lot for really telling people what kind of game they're going to be getting into most of the time.
Best Answer
There is no difference between a module and an adventure. Module is simply an archaic term for adventure.
Early TSR D&D/AD&D adventure products called themselves "modules," but in their covers and text made no distinction between an "adventure", a "module," and an "adventure module," they used the term interchangeably.
Here's a picture of X1 Isle of Dread from 1981. The header claims it's "Dungeon Module X1." The subheader declares it "A Wilderness Adventure." The text says "first in a series of adventure modules." The oval at the bottom calls it a "Special Introductory Wilderness Module."
"Books - Modules - Miniatures, the Dragonlance Saga Continues," proclaims the back of DL3 Dragons of Hope.
Other publishers didn't adopt the "module" nomenclature as much. On my various non-licensed Mayfair Role Aids AD&D adventures and sourcebooks from the 1980s, they studiously just call themselves "adventures."
But are modules setting independent?
No.
Many were, of course, with the idea that you could use them in your own campaign world. But there were many modules explicitly set in Greyhawk or another published game world. T1-4 Temple of Elemental Evil proclaims on its cover "A Monstrous Adventure! Includes Tl , The Village of Hommlett and the long awaited T2, Temple of Elemental Evil set in the WORLD OF GREYHAWK Setting." There was also a WG series of modules that was not just set in Greyhawk but got deeply into the locations, NPCs, and metaplots of the setting beginning in 1982.
But are modules more - "modular", and not full campaigns like an Adventure Path is?
Not by the 1980s.
There was no term "Adventure Path" back then, but when they put together larger arcs of adventures they were still called modules. Take the plot arcs like I3-I5, Desert of Desolation series. When republished, the foreword talks about "[the authors] provided an excellent set of adventures in the original Desert of Desolation Series, I3 through I5. The story that held these adventures together was of a campaign style, an epic quest against seemingly insuperable odds." It was recognized that you could have series of adventure modules that did provide an entire campaign.
Take D3 Vault of the Drow (1980) - its cover says "This module can be played alone, as the conclusion to module D1-2 (Descent Into The Depths Of The Earth), or as the third module in a series that forms a special extended adventure (G1-2-3 Against the Giants, D1-2, and Q1 Queen of the Demonweb Pits). (Eventually it was republished as GDQ1-7 as one large softcover.)
Or T1-4, Temple of Elemental Evil, which when published in one larger softcover, its cover refers to it as "Campaign Adventure," but then in the Introduction starts in with "This module is designed for..." Modules could have plot and be interlinked and have many parts.
In fact, I can't find the term on any of the products themselves, but when the original combination modules (T1-4, A1-4, GDQ1-7, I3-5) came out, we called them "supermodules." In fact, that has to be somewhat official because it's in the product name on Amazon!
TSR continued to (somewhat intermittently) use the term “module” through 2e in the 1990s, but left the terminology behind with the advent of 3e.
TL;DR
Basically, "module" was a term TSR started using early in D&D, probably because earlier wargames had "expansion modules", and it stuck for a decade, but then gave way to the more readily understandable "adventure", especially as other publishers who didn't start with that term published games and supplements.
All "well to me adventure means this and module means this" statements are basically made up out of people's minds, except inasmuch as "module" reminds one of 1970s-1990s D&D.
Nowadays, people use "adventure" and "adventure path" as common terms, but really Paizo coined the "adventure path" nomenclature in the late days of Dungeon, before then you just had "campaigns" or "campaign adventures."