You are correct. It is precisely this ability to map seemingly arbitrary and meaningless physical manifestations of turning machines to the abstract that provi

You are correct. It is precisely this ability to map seemingly arbitrary and meaningless physical manifestations of turning machines to the abstract that provides the concept so much power. By being able to provably map the implementation to the concept you're able to get all the "benefits" of the concept without having to individually prove them in your implementation where it might be really difficult or just time consuming to prove everything individually. This, coincidentally, is the same reason mathematics is so obsessed with isomorphisms, homeomorphisms, homomorphisms etc. They are all different ways of saying "this thing stays the same, even when it changes" which many times means that things proven in a place where it's easy to prove are maintained in places where it's very difficult or even impossible to prove (absent the isomorphism).