There's a good case to be made that the information-theoretic definition of entropy is the most fundamental one, and the version that shows up in physics is just that concept as applied to physics.
My favorite course I took as part of my physics degree was statistical mechanics. It leaned way closer to information theory than I would have expected going in, but in retrospect should have been obvious.
Unrelated: my favorite bit from any physics book is probably still the introduction of the first chapter of "States of Matter" by David Goodstein: "Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics."
Not really. Information theory applies to anything probability applies to, including many situations that aren't "physics" per se. For instance it has a lot to do with algorithms and data as well. I think of it as being at the level of geometry and calculus.
Yeah, people seemingly misunderstand that the entropy applied to thermodynamics is simply an aggregate statistic that summarizes the complex state of the thermodynamic system as a single real number.
The fact that entropy always rises etc, has nothing to do with the statistical concept of entropy itself. It simply is an easier way to express the physics concept that individual atoms spread out their kinetic energy across a large volume.
I'm not sure that's quite the right perspective. It's not a coincidence that entropy increases over time; the increase in entropy seems to be very fundamental to the way physics goes. I prefer the interpretation "physics doesn't care what direction the arrow of time points, but we perceive it as pointing in the direction of increasing entropy". Although that's not totally satisfying either.
Unrelated: my favorite bit from any physics book is probably still the introduction of the first chapter of "States of Matter" by David Goodstein: "Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics."