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.
There is no generic edition of Apocalypse World — that is, an edition with the setting stripped out and “just” the rules.
This is for the simple reason that the rules effectively are the setting, so there's no way to have “just” the rules. All the other Powered by the Apocalypse games were created by playing and studying Apocalypse World (and other PbtA games) to understand the design patterns involved, and then adjusting the rules to create rules that embodied a new setting.
However, one of the most well-respected designers of PbtA games — Avery Alder of Monsterhearts fame — has created a free PDF meta-game called Simple World (alternative link due to today's server issues) that is kind of like a playbook for a PbtA game. You follow the instructions in Simple World, and the result is a GM's Agenda, Principles, and Moves that are customised for the setting and themes you want to play with. It meanwhile instructs the players in how to create characters that suit this custom set of rules. This is as close to a generic set of PbtA rules that exist. Creating a game is fast (you do it together as a group while the players make their characters), so you can create a custom PbtA game and start playing in one session.
And once you've created a PbtA game with Simple World, you can always continue to adjust, tweak, and refine it, along with formalising some iconic character playbooks, ending up with a true stand-alone PbtA game of your very own design.
Best Answer
In common parlance, a 'Powered by the Apocalypse' game is a game that uses the same resolution engine of Apocalypse World, by Vincent D. Baker, or is somehow recognizably inspired by or similar to it.
In other word, it is a way for people to categorize Apocalypse World hacks.
Despite this being the perceived definition, commonly used in a large part of the Internet including the tag description in this site and Wikipedia, PbtA is in reality something different.
Vincent and Meguey Baker stated that: