Out of the above examples, none of the ones I'm familiar with are true dei ex machina. Yeah, that's right, I declined the Latin noun correctly. Or at least attempted to, it's been a long time since I took Latin. If that strikes you as pretentious, probably don't read the rest of this post.
The OP correctly defines a deus ex machina as a "plot device whereby a seemingly inextricable problem is suddenly and abruptly solved with the contrived and unexpected intervention of some new event, character, ability, or object.)"
The HL2 gravity gun surviving disintegration and becoming superpowered doesn't fit as a deus ex machina. Yeah, it takes on new and unexpected powers, but we've been told that the gravity gun was exceptional and revolutionary. And metanarratively, we know the GG is central to the narrative and mechanics because Valve is basically saying "physics OMG LOL" the whole time, and teaching us how to use it in myriad situations and against myriad enemies.
The action movie cliche of a hero succeeding because he's angered by the death of a loved one doesn't qualify either. It's certainly implausible, sappy, and narratively weak - but it's not a deus ex machina. The character and his/her relationship to the hero appeared beforehand, as did the villain's malevolent/murderous tendencies. A "twist", or a new plot development, is not immediately a deus ex machina, even if it feels unearned.
Same goes for James Bond's gadgets - they're introduced well before they're used, and we're made familiar with their properties. It's certainly contrived that the precise situations where these gadgets are most useful emerge so conveniently, but not every contrivance is a deus ex machina.
So what is a deus ex machina? The best example I can think of is the end of Saving Private Ryan. As a wounded Tom Hanks is ineffectually shooting at an advancing tank with his pistol, it explodes. For one dumb half-second, we wonder if a bullet found some ***** in the armor, igniting the tank's ammunition or fuel supply by some freak ricochet. And then a P-51 Mustang soars through the smoke and flame. A jubilant soldier calls the planes "angels on our shoulders."
As dei ex machina go, this actually works pretty damn well. At no point during the final battle were we told that the soldiers could expect relief or air support - we actually get the opposite message. But throughout the movie, Spielberg has told us that war is chaos and randomness. In the opening Omaha Beach scene, whole PT boats get slaughtered before the soldiers even make landfall. One of those men could have been the best soldier in the entire war, but it didn't matter. Sometimes you're just in the wrong place at the wrong time. But sometimes, you get lucky. So Spielberg pulls it off.
