There is no re-definition of "arrow of time" going on here. Shalizi is using the phrase in its standard thermodynamic sense, describing the fact that a number of macroscopic processes are thermodynamically irreversible.

Consider a specific example: two boxes of gas initially at different temperatures are brought into contact through a diathermal barrier. I check the temperature of these gases using a thermometer periodically. I observe that over time temperature difference vanishes. The gases spontaneously equilibriate.

What would you say about what's going on here? The standard story is that the thermal equilibriation takes place due to the Second Law of Thermodynamics. Heat transfer from the hotter gas to the colder one leads to entropy increase. From a (Boltzmannian) statistical mechanical perspective, the region of phase space corresponding to both gases having the same temperature is larger than the region of phase space corresponding to them having different temperatures. So a distribution that is uniform over the former region (and vanishes elsewhere) will have a higher entropy than a distribution that is uniform over the latter region. Note that none of this requires any appeal to the entropy associated with the observer. The entropy increase in this case has nothing to do with the observer's memory. It has to do with heat flowing from one box of gas to the other.

Now it seems like the guy you link to objects that we can't say that the Second Law applies to this two-gas system because the system is not completely isolated. But this ignores two things. First, the Second Law has productively been used a huge number of times in the past to describe the behavior of systems exactly like this. Second, by this standard the Second Law does not apply to any system. There is no actual system that is completely isolated, except the universe as a whole.

The thing is, the two-gas system is "isolated enough". There is no significant mechanical work being performed on or by it (discounting the negligible amount of work required to raise the mercury in the thermometer I use), as there is in the case of a refrigerator. Observing a system's state does involve some exchange of energy with it, but it need not involve doing work on the system.

Now, Shalizi's point is that if we are strict Bayesians about the state of the system, then the entropy of the distribution we associate with it will not increase, so we would say that the entropy of the system is decreasing. But this is wrong! The entropy of the system is increasing. Not the entropy of the system+observer combo, the entropy of the system itself. If your approach to statistical mechanics tells you it is not, then you are the one flying in the face of orthodox thermodynamics, not Shalizi.