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.
I have done some game-design work, mostly in D&D 3.5e-derived settings, which are roll-over. However, some roll-under ideas were considered—albeit briefly—for some of these mechanics, so I’ve given this some thought and discussed these with other designers. I cannot claim as much experience with this question, though, as someone who has done a lot of work with roll-over, roll-under, and mixed systems. But from my perspective,
There is no particular “advantage” to mixing the two; that is pure downside, because it makes the game less consistent and forces players to remember which rolls are which.
But picking one and enforcing consistency has its own downside: there can be advantages to one or the other in certain situations, but if you enforce consistency you cannot use the “better” approach when it would make sense. For examples:
Roll-under has the nice property for percentile rolls that your target number is also your chance of success: if you must roll a 20 or less on a d100, you have a 20% chance of succeeding. For roll-over, the same 20% chance would be a roll above 80—you have to do 100−x to determine your odds each time.
On the other hand, roll-over works much better for unbounded numbers: you can always increase a target number in roll-over, but decreasing a target number in roll-under is somewhat awkward when it gets negative.
In short, by not having the game consistently use one or the other, you are free to pick whichever is appropriate for a given roll, not being constrained by the game using the other type for everything else. The downside is, the freer you are with this—the more you use whatever roll type seems most appropriate for each roll, the harder it’s going to be for players to remember which rolls are which.
Best Answer
You can also encourage everyone at the table to follow guidelines established by improv practitioners; these become less "rules of the game" and more "rules of play" (i.e. they don't really answer "what can I do next" as much as they address "whatever I'm going to do next, how do I do it?"):
Accept every offer. During the course of play, other players will make suggestions about the ongoing narrative, sometimes directly involving your character. Accept what's offered; don't block their suggestions. Try, as much as possible not to say "no", literally, or figuratively. Don't say "no", say "yes".
Accept and build with new offers. During the course of player, when you have an opportunity for agreement, don't just agree (see first guideline). Take what's offered, and build on it with a return offer. There are two ways to do this: add detail, and add complication. Note that both of these help you steer around the natural reaction against the first guideline ("I don't want that!") to "get what you want" or "avoid what you don't want" for your character. Don't just say "yes", say "yes, and..." or "yes, but...".
Make simple offers. There are a bunch of other players at the table; just as with boardgames, play rolls along more smoothly when there's as little down-time as possible. Engaging the other players is key, so when you provide details, be simple, straightforward, and incremental and then throw the tempo to another player (or the referee). When you say "yes, and..." (or "yes, but...") say only one thing.
These are also commonly known as "No Blocking", "No Wimping", and "No Steamrolling".
For some more explanation about how these things work in improvisational theatre, check out the Improv 101 series of blog posts (I found them by googling "no blocking no wimping", so they were a fast find, not necessarily "the best" find -- lots of other improv resources exist). There are lots of ways in which improv doesn't map well to RPG gaming, but in a lot of ways, good advice there is good advice here.
Based on personal experience, I might suggest one that's gaming specific and in the spirit of the improv guidelines:
The game is a player. You can think of the game's rule engine itself as a player at the table: so, when it has offers (i.e. you use the rules to help adjudicate an outcome), don't block or wimp or steamroll its offers. As much as you possibly can, resist temptation to fudge the offers of the game itself; as much as you possibly can, try to play the game as written (before you decide some part of it is "clearly broken" or "not to your taste"). In Burning Wheel, terms, you can think of this as "Say yes, and roll the dice..." The dice can suck, but you must give them that freedom: it makes for memorable play experience.