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.
The Judge's Shield (1977) predates the Dungeon Master's Screen by about two years, give or take some months. (That the Judge's Guild got to this first is not surprising, as Gygax infamously told them that nothing but rules would sell, and the Judge's Guild was fated to prove him wrong.) So the term “shield” was established before the term “screen”, and still has minority currency today.
The Referee Screen (1979) for Traveller has the same vintage as the DM's Screen and is the only other data point before the 80s begin. It's unclear which was published first that year and therefore which one coined the term “screen”, but together with their two popular games — D&D and Traveller — they certainly had a large enough combined audience to establish the term “screen” in common usage over the the term “shield”, and likely TSR's audience alone would have been enough to establish the term as definitive. (Judge's Guild material, although designed for use with D&D and acclaimed at the time, was not “official” and that would have prevented a lot of play groups that didn't stray from TSR-branded material from being exposed to the alternate term.)
The history isn't very interesting, unfortunately. The only reason we have more than one word for it is because “shield” enjoyed first-to-market advantage in establishing mindshare enough to not be completely destroyed by the TSR juggernaut's adoption and promulgation of the term “screen.”
The “GM Screen” variation, as noted in the other question's answer, is merely a variation of the term that follows the separate trend of the hobby adopting “game master” as a non-trademarked, non-D&D-specific term, whether out of a hobbyist's need for precision or a publisher' need to avoid legal trouble. The term “GM screen” therefore doesn't have much of an interesting history of its own. You can see “GM screens” in a few early items:
Best Answer
Game Master (GM) is a generic term for the person who's running the game.
Some games have their own name for that role — Dungeon Master is the term Dungeons & Dragons uses. It's even a Wizards of the Coast trademark. (Yes, really.)
Some games use “GM” for their actual official name for the role, such as Fate or Dungeon World. Others use a more specific term. We can often use “GM” as a generic replacement for this type of role regardless. Games vary considerably in the roles and responsibilities they place upon the GM, so one game's GM is not necessarily another's, and some games have chosen their term for this role (such as “Master of Ceremonies” in Apocalypse World) specifically to evoke a reminder of those responsibilities.
We didn't invent these terms. Dungeon Master arose sometime in the 70's, and Game Master was already in use before our hobby started: see inky's chronicle of the history of these terms here.