I've spent my entire life around dogs - and that dog? As others have said? Left the mark it intended to. If it wanted to bite into your nephew's neck (as horrific a prospect as that actually is)... then it would have done so, regardless of how blunted by age it's teeth would be at 10 years. Have you ever tried to get a dog's jaw unlocked when it's determined? They're ridiculously strong when locked around... anything. The straight legged bounce it's making looks like it thinks it's joining in with the game, in my opinion - but the sad truth of it is, that it's an animal which shouldn't be joining in with small children's games. Because whether it means to or not, it'll hurt them - be that accidentally or deliberately (because a dog can forget that children are more fragile than puppies/other dogs happen to be).
I have a 13 year old Springer Spaniel who is incredibly tetchy around children - except for my 14 year old, whom he grew up with. He barely puts up with their oddities, and will politely hold himself aloof from them at all times. He was badly teased by a neighbour's children when he was a puppy, though, which is why he does this. I also had to peel toddlers from around his neck and prise their fingers out of his ears when out walking him, on a regular basis - he wouldn't growl at them, though, he'd simply look at me as if to say "I appear to have acquired a limpet, Mother... remove it for me, please!". Had he made any signs of wanting to remove a toddler himself, though... he would have to be destroyed. That's my rule of thumb, I'm afraid, despite the fact that I know my dogs are trained, would only bite if they felt they had no other choice in order to defend themselves, and I would have failed them by allowing the situation to develop in the first place. A child's skin is infinitely thinner than a dog's is - with good reason. A dog needs thick skin to prevent serious injury in fights for dominance. Which no matter how well trained/soppy/daft around kids a dog is... happen. A lot. Even my own two dogs will clash from time to time - usually in very hot weather, or if the younger one is pushing boundaries and the older one is putting them firmly back into place. The term "top dog" exists for a reason.
Your cousin ought to know her own dog's body language - and the fact that this escalated, however quickly, to the stage where its teeth came into contact with a small child's neck... is proof that she doesn't. She needs to retrain herself as a dog owner - and accept responsibility for the fact that she let her dog down, just as much as it's let her down by doing this.
I have a smaller dog, who gets very stressed in certain situations. So I don't put her in those situations, in the same way that I keep my older dog away from other people's children (to the point where I've turned down invitations to family parties where I know there will be children, despite "oh, bring the dog... he'll be fine!" - because I respect his need for peace and not having toddlers clamped around his neck). She doesn't like loud noises, people running, and small dogs (which is ludicrous, really, because she is a small dog!). When we're out, if she recognises any of her stress factors, she will position herself behind me. I become a barrier between her and whatever it is that's frightened her. It alerts me to the fact that she's stressed, and I make sure that her fear isn't elevated any further - ie, to the point where she'd do what your cousin's dog did. They're animals. And size doesn't matter, they all have very powerful jaws which once they clamp onto something? Are nigh on impossible for a human, however panicked, to prise open again. And the more a dog smells fear, the worse it's own primal instincts become. Whilst I completely understand your BIL's panic at what happened, it could have caused the dog to clamp down on your nephew's body - neck, arm, leg - hard enough to puncture flesh... which no one - not even your cousin - wants.
There doesn't have to be a family rift over this - although there will be/already is. Your cousin simply needs to shut her dog away if there are children visiting. A crate, or in another room that children are told to stay out of. Even a 2 year old is capable of understanding that they need to respect a dog's personal space, if an adult explains to them that the dog is old, and a bit grumpy, and wants to sleep rather than get excited by the child's play. And if they're not capable of learning to respect another being's space/needs? Then the child needs to be removed until it is. For their own safety. Either way, the adults need to own their own responsibilities in this situation. Your cousin needs to apologise (profusely) for her dog's behaviour - and think about why it reacted in the way it did (is it in pain, were the screams of playing children too loud for its ears, was it over-stimulated/frightened/stressed by a crowd of people, was it trying to tell the children to stop shrieking...) and not have it roaming around at family parties again. Your sister and BIL need to explain to their children that dogs are animals, and will react accordingly. Your nephew didn't do anything wrong. He's a child. But his parents are just as culpable as your cousin is for allowing this situation to develop in the first place. The dog didn't want the noise/children running around in its home, it reacted in an effort to reclaim its space/some peace, now everyone's blaming it... not the adults who should have recognised its agitation and prevented your nephew from being in a situation where yes; he could have been seriously hurt.
Please note I said "could have been" - because other than a few red marks, and a scare because his father panicked, he's fine (although he might have a phobia about dogs brewing as a result...). Many years ago, my then 2 year old cousin tripped over a pair of shoes and landed on our grandparents elderly, sleeping Corgi... which reacted instinctively to having been woken from her nap, and hurt by the toddler falling onto her. She had a snaggle-tooth which jutted out of her mouth... and as she lifted her head, mid-growl, it caught against the skin beneath my cousin's eye - leaving a red mark. My aunt and uncle went ballistic - because the dog had growled and her snaggle-tooth had marked their baby's face. They demanded that my grandparents have "the vicious dog" destroyed immediately... or they'd never allow their (precious first-born) child to visit with them again. My cousin had been laughing up until the moment his parents started to shout, and grab at him, and point at the small mark on his face (just under his eye... an inch higher, it would have caused an injury, so I do understand their terror) - at which point, he burst into tears and was inconsolable to the point where he began to retch. More shouting about "that dangerous dog" and "look what she's done!" followed, because my cousin was terrified. That was 40 years ago. My grandparents had no contact with my aunt, uncle and cousins for, I think, ten years or so. They didn't get to see my younger cousins as babies or toddlers at all, and consequently, had a very strained relationship with their own grandchildren - because there wasn't the same bond between them, as there was with the rest of us. I know it broke my grandmother's heart, because essentially she lost her bond with her youngest son, too, as well as with her first grandson. The "vicious", "dangerous" Corgi? In the immediate aftermath, she was shut away whenever children visited (which I suspect she preferred), and she lived for another four or five years. She reacted to being suddenly landed on and woken from a nap, in her own home, where she always napped - and the snaggle-tooth just happened to mark my cousin's skin. It was purely accidental on both their parts - him for falling on her, her for growling as she was woken up by him landing on her. Even as an 8 year old I knew that. But my cousin? Has been terrified of dogs ever since - and yes; I do blame my aunt and uncle for this, in so much as their panic frightened him, and he associated his panic with the dog, ergo, in his mind, all dogs are something to be frightened of/panicked by. He cannot allow his own children to be anywhere near dogs, as a result, which means that they associate dogs with their dad being panicky, therefore dogs are something for them to be frightened of/by, too. When we meet up, I leave my dogs at home - because I respect my cousin's fear, don't want him or his children to be frightened, won't put my dogs in a situation where they'd pick up on the fear and react to the stimulas by possibly snapping, or growling, which would prove my cousin "right" in that all dogs are dangerous... Which they are. Of course they are. They're animals which I am responsible for - and, as such, I won't knowingly, deliberately put them into a situation where they might hurt someone, accidentally or not.
Keep the dog away from children. Visit with your cousin outside of her home, without the dog. Family harmony can be restored - if your cousin and the child's parents/grandparents all accept their responsibility for this situation ever having been allowed to develop in the first place... and go out of their ways to prevent it from happening again.