GitHub specifically invented the idea of "forking" as a social action on a forge site that allows you to create your own associated copy of a repository. This is related to but different from the broader meaning of "fork". "Fork" doesn't mean anything at the git level.
I'm not sure exactly the distinction that you're trying to make. I see GitHub's use of "fork" as a specific application of the broader meaning of "fork", not an invention of a new and distinct concept. Just as putting "wheels" onto a steam engine can produce a new type of vehicle but doesn't change the concept of "wheels", GitHub's use of "fork" doesn't fundamentally change the broader concept of "fork".
If any changes done to the parent repository propagate automatically to "forked" repositories without the explicit consent of the _owner_ of the fork then it does change the broader concept of fork, and to follow your analogy it would be like calling a caterpillar track a wheel.
If this is acceptable because the original version it's a private repository that is unrelated, what we are discussing is the meaning of the word itself.
I think we are in agreement. Because access to the "forked" repository was removed without the consent of the owner of the fork, it is inaccurate for GitHub to describe it as a "fork". For clarity, I would also describe the "owner" of the fork as the person who created the fork.
