History

The reason for the set mix as it exists is that, originally, the dice available were a set of platonic solids, sold by an educational company and repurposed by TSR. Namely, a tetrahedron (d4), cube (square hexahedron, d6), equilateral octohedron (d8), dodecahedron (d12), icosahedron (d20). This was a "platonic solids" set.

D20's were routinely read as d10's in wargaming, and used for generating percentiles, which made use of the d20 as a d10 or pair as a d100 a standard practice, going back as early as 1972.

A 1938 dodecahedron with 2 rounded faces opposite each other was available in 1938 for stock simulations. ➀ It wasn't used by TSR, but shows the probable origins of the pentagonal dipyramid we now think of as the d10.

Names

As to names - the earliest modern set (with dipyramid d10) I can readily find documented by name is the "Dragon Dice" set from 1981 (as evidenced on the packaging copyright notice), as photographed at Dice Collector. ➁

I can, however, cite TSR "catalogue" from a ©1979 TSR product - Swords & Spells lists the older set as "Multi-Sided Dice Set" in the catalogue extract on the back flyleaf. ➂ It reads:

Multi-Sided Dice Sets — Each set contains one 20-, 12-, 8-, 6-, and 4-sided die

The same title and text was used in the 1975 product list in Strategic Review Vol 1. Issue 3 (1975). ➃

So:
Multi-Sided Dice Set - the original d20 d12 d8 d6 d4 platonic set.
Dragon Dice - the "mud dice" set with d10's.

➀ Dice Collector - Mason & Co Stock Exchange Dice - www.dicecollector.com/MINT39_MASON_&_CO_STOCK_EXCHANGE_DICE.jpg

➁ Dice Collector - TSR - www.dicecollector.com/THE_DICE_THEME_TSR.html

➂ Gygax, [E.] Gary, Swords & Spells, 6th printing, Tactical Studies Rules, 1979. Original copyright 1976.

➃ Tactical Studies Rules, Strategic Review, Vol. 1, No. 3, Autumn 1975, Ed. [E.] Gary Gygax. TSR advert on page 8. From the Dragon Archive CD.

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.