I feel like this may be a semantics issue. I think that order implies information. To me, saying that a system becomes more ordered implies that I know about the increased order somehow. Under that construction, disorder (i.e. the absence of detectable patterns) is a measure of ignorance and disorder then is closely related to entropy. You may be preserving a distinction between the map and territory (i.e. between the system and our knowledge of the system) that I'm neglecting. I'm not sure which framework is more useful/productive.
I think it's definitely an important distinction to be aware of either way.
'order' is not a well-defined concept. One person's order is another's chaos. Entropy, on the other hand, is a well-defined concept.
Even though entropy depends on the information you have about the system, the way that it depends on that is not subjective, and any two observers with the same amount of information about the system must come up with the exact same quantity for entropy.
All of this might seem counter-intuitive at first but it makes sense when you realize that Entropy(system) isn't well-defined, but Entropy(system, model) is precisely defined. The 'model' is what Bayesians would call the prior. It is always there, either implicitly or explicitly.
Sean Carroll et al. posted a preprint with the above title. Sean also has a discussion of it in his blog.
While I am a physicist by training, statistical mechanics and thermodynamics is not my strong suit, and I hope someone with expertise in the area can give their perspective on the paper. For now, here is my summary, apologies for any potential errors:
There is a tension between different definitions of entropy: Boltzmann entropy, which counts macroscopically indistinguishable microstates always increases, except for extremely rare decreases. Gibbs/Shannon entropy, which counts our knowledge of a system, can decrease if an observer examines the system and learns something new about it. Jaynes had a paper on that topic, Eliezer discussed this in the Sequences, and spxtr recently wrote a post about it. Now Carroll and collaborators propose the "Bayesian Second Law" that quantifies this decrease in Gibbs/Shannon entropy due to a measurement:
[...] we derive the Bayesian Second Law of Thermodynamics, which relates the original (un-updated) distribution at initial and final times to the updated distribution at initial and final times. That relationship makes use of the cross entropy between two distributions [...]
[...] the Bayesian Second Law (BSL) tells us that this lack of knowledge — the amount we would learn on average by being told the exact state of the system, given that we were using the un-updated distribution — is always larger at the end of the experiment than at the beginning (up to corrections because the system may be emitting heat)
This last point seems to resolve the tension between the two definitions of entropy, and has applications to non-equilibrium processes, where an observer is replaced with an outcome of some natural process, such as RNA self-assembly.