Entropy in physics is usually the Shannon entropy of the probability distribution over system microstates given known temperature and pressure. If the system is in equilibrium then this is objective.
That's not a problem, as the GP's post is trying to state a mathematical relation not a historical attribution. Often newer concepts shed light on older ones. As Baez's article says, Gibbs entropy is Shannon's entropy of an associated distribution(multiplied by the constant k).
It is a problem because all three come with a bagage. Almost none of the things discussed in this thread are invalid when discussing actual physical entropy even though the equations are superficially similar. And then there are lots of people being confidently wrong because they assume that it’s just one concept. It really is not.
Don't see how the connection is superficial. Even the classical macroscopic definition of entropy as ΔS=∫TdQ can be derived from the information theory perspective as Baez shows in article(using entropy maximizing distributions and Lagrange multipliers). If you have a more specific critique, it would be good to discuss.
In classical physics there is no real objective randomness. Particles have a defined position and momentum and those evolve deterministically. If you somehow learned these then the shannon entropy is zero. If entropy is zero then all kinds of things break down.
So now you are forced to consider e.g. temperature an impossibility without quantum-derived randomness, even though temperature does not really seem to be a quantum thing.
If information-theoretical and statistical mechanics entropies are NOT the same (or at least, deeply connected) then what stops us from having a little guy[0] sort all the particles in a gas to extract more energy from them?
Sounds like a non-sequitur to me; what are you implying about the Maxwell's demon thought experiment vs the comparison between Shannon and stat-mech entropy?