The limits of natural selection is nature; what you're describing in the two examples you propose is what we call artificial selection, not natural selection.
The difference? Two moths land on a white piece of granite. One is white, the other is black. A bird sees the black one and swoops down and eats it. That's natural selection, and most of the moths around white rocks will tend towards being white.
On the other hand, a small tribe of humans sits around their campfire at night, and it attracts wolves who can smell the meat cooking. Some of those wolves are friendly to the humans, and they feed them. Others are not, and get nothing. In time, humans restrict the breeding of wolves to those who are most friendly, instantly kill any animal that shows aggression in any form, and rewards actions like protecting the camp area from other animals, or alerting the humans to threats.
Dogs are born. From artificial selection, or selective breeding.
In the case of Stephen Hawking, his survival was as much a testament to his own will as modern technology. Most people with Motor Neuron Disease are lucky to survive 5 years after diagnosis, let alone close to fifty. But, to be fair, we as a species are artificially selecting ourselves as a whole, ensuring the survival of those nature would deem fit as well as (in many cases) those it would deem unfit. This is still a form of artificial selection because we choose to do it.
Your one-world government is doing exactly the same. They're selectively breeding the next generation, and that's artificial selection, not natural selection. There are those who argue that a version of this has already been done with humans, pointing to slave owners in the USA prior to the Civil War. They certainly rewarded size, strength, agility, and other physical traits by pairing up such people with their similar from the other gender, encouraging children. This was obviously not morally sound, but it is a clearly documented example of human selective breeding from our history.
Ultimately, the differentiation is intent, or lack thereof. There is material out there about the increasing divergence between egg chickens and meat chickens in terms of their unity as a species (I won't link it because it can cause distress) but this is yet another case of humans intentionally encouraging a specific trait in a specific set of animals. That, however, is not natural selection as the term was originally intended to be used by Darwin.