Credentials aren't very relevant here, but if we're going to talk about them, I have a PhD in engineering and a BS in math (minor in physics).

and in a way which is no more subjective than the "spooky action at a distance" of electromagnetic and gravitational fields.

Again, as I've pointed out at least once before, entropy is not subjective. Being dependent on model and information does not mean it is subjective.

And so the entropy at this energy of this isolated system is log(N(E)) where N(E) is the number of states that have energy E.

Right off the bat, this is wrong. In a continuous system the state space could be continuous (uncountably infinite) and so N(E) makes no sense. "Logarithm of the number of states of the system" is just a loose way of describing what entropy is, not a precise way.

and so the number of states the system can be in is 1 because you KNOW which one it must be in. And so its entropy is 0.

The number of states a system can be in is always 1! A system (a classical system, at least) can never be in more than one state at a time. The 'number of states', insofar as it is loosely used, means the size of the state space according to our model and our information about the system.

And its entropy is log(N(E)), the count of states it evolves through, and it is unchanged that you know at each instant which state it is in.

There are several things wrong with this. First of all, it assumes the ergodic hypothesis (time average = space average) and the ergodic hypothesis is not required for thermodynamics to work (although it does make a lot of physical systems easier to analyze). But it also has another problem in that it makes entropy dependent on time scale. That is, choosing a fine time scale would decrease entropy. This is not how entropy works. And at any rate, it's not what entropy measures anyway.

I said rotation or movement of a rigid body. By definition a rigid body doesn't have modes of vibration in it.

But I'm not assuming a rigid body. You are. There is no reason to assume a rigid body. I offered an example of a cold flywheel turning a hot flywheel, as a system where energy moves from a cold object to a hot object. You decided for some reason that the flywheels must be rigid bodies. They aren't, at least not in my example.