Why does time not run backwards inside a refrigerator?
The short answer is that the entropy in the system decreases only due to the fact that the outflux of entropy is larger than the local growth. But the positive local growth of entropy is what is important for the arrow of time and that is why time is not running backwards in a refrigerator.
This can be demonstrated by using a fluid description of the open system (refrigerator). In fluids, the local increase of entropy can be expressed by the non-conservation of the entropy density $\sigma$: $$\frac{\partial \sigma}{\partial t} + \nabla \cdot (\sigma \vec{v}) \geq 0 $$ where $\sigma\equiv \Delta S /\Delta V$ is the amount of entropy $\Delta S$ in an infinitesimal volume $\Delta V$ at a given point.
You can see that the inequality above is not symmetric with respect to time reversal (which leads to a minus in front of $\partial/\partial t$ and $\vec{v} = d \vec{x}/dt$ ). Its meaning is exactly a local formulation of the law of increase of entropy. I.e., as long as the law above is fulfilled, the arrow of time is running correctly and in the right direction. We will see in the following that a decrease of entropy of a larger open system is not in conflict with this local time arrow.
Let us now study our open system of volume $V$ with a boundary surface $\Sigma$. We integrate the inequality above over this whole volume to obtain $$ -\frac{\partial}{\partial t} \int_V\sigma d V \leq \int_V \nabla \cdot(\sigma \vec{v}) dV $$ The integral of entropy density over the volume of the system is of course simply the total entropy in the system $S_{tot}$. The left-hand side of this new integral inequality is then simply the decrease of total entropy of our system. Also, we can take the right-hand side and use the Gauss (or Divergence) theorem to express it as an integral only over the surface of our open system $$ \int_V \nabla \cdot(\sigma \vec{v}) dV = \int_\Sigma \sigma \vec{v} \cdot d \vec{\Sigma} $$ Physically, $\sigma \vec{v} \cdot d \vec{\Sigma}$ is the flux of entropy outside the system through an infinitesimal element of the boundary surface of the system $\Sigma$. The whole integral is then simply the total flux of entropy out of our open system.
We have thus derived an inequality $$-\frac{\partial S_{tot}}{\partial t} \leq \int_\Sigma \sigma \vec{v} \cdot d \vec{\Sigma} $$ The left-hand side is the total decrease of entropy in our system, and the right-hand side is the total flux of entropy out of the system. Now, you can see explicitly that a decrease of entropy of an open system is entirely consistent with the local law of entropy increase as long as the amount of entropy leaving the system is larger than the amount of entropy decreased. This is also the case of the fridge and any cooling system.
But should that also mean, that no arrow of time can be defined for an open system like a refrigerator? Or must we conclude that the connection between entropy and time is an illusion? If we cannot use entropy to define an arrow of time inside an open system, what is it that makes sure that time doesn't run backwards inside an refrigerator?
Let's take a scientist who lives in our world. After some time experimenting around, he will notice the following thing:
If two bodies are put into physical contact, energy will always spontaneously flow from the hotter body to the colder body, and never in the opposite direction.
Here, the key word is "spontaneously", without any work having been done.
The scientist will then define the "forward" direction in time as the direction in which energy flows spontaneously from an hotter body to a colder body.
Let's now consider a mini-scientist living inside the fridge. He was born inside it, and doesn't know any reality external from the fridge. He will never see heat flow from the inside to the outside, because there is no "outside" for him (let's make the hypothesis that temperature is kept approximately constant inside the fridge). How will the mini-scientist define the "forward" direction in time?
The answer is: in the same way as the scientist who lives outside. Indeed, if two objects at different temperature are put into contact inside the fridge, heat will always flow spontaneously from the hotter body to the colder body (and, like you stated, the pieces of a broken glass will not magically come back together just because we are inside a fridge!).
Yes, an outside observer will see that there is an heat flow from the colder inside of the fridge to the hotter outside environment, but he will also see that the fridge is plugged in and that work is being done, so the process is not spontaneous and the definition of the "forward" direction in time is safe.
So I would say that the problem is only apparent, and that there are no problems when defining the arrow of time inside an open system.
The arrow of time in a thermodynamic system should be thought instead as a statement about time-reversal invariance. For example, in classical mechanics, based solely on the particle motion one cannot tell if time is going forward or backward. In the same way, in an open system one cannot tell which way time is flowing based on the change in entropy alone. In essence, entropy does not define the direction of time in an open system.
However, this is different from asking about why time doesn't run backwards in a refrigerator. If you think of a fridge as something that decreases entropy over time, one can think of its time-reversed partner as a heater which increases entropy over time. Hence, in thinking of the system as a refrigerator, you've already selected a particular time direction. Nobody is stopping you from defining time to flow in the opposite direction, but asking why time doesn't run backwards in a fridge isn't really a well-posed question. Saying time flows backwards is a relative statement; you have to tell me what "backwards" is relative to.
EDIT: My answer seems to be unsatisfactory, so let me try to elaborate.
First, what is meant by time? It is a 1D parameter that controls properties of objects (eg position). You can think of each physical property (object) independently having their own time parameter.
Now, it is interesting to note that the microscopic laws of physics are time-reversal invariant. This means that the form of the equations don't change from $t\rightarrow -t$. However, you'll still get qualitative differences, such as velocity $v \rightarrow -v$ under time-reversal. Hence, a particle moving to the right will be moving to the left under time reversal. Hence, what time-reversal invariance means is that if you look at a clip of a particle moving forward in time vs moving backward in time, you can't tell which one is which. So, if we have don't know which way time should be going, we can just pick one of our choosing.
Then, we can just choose a time direction for every single particle in a collection however we want. But wait a minute; if I watch all the particles move, then clearly their time directions should all be synchronized to my time. Hence, we stumble upon a key property of time that it depends on the observer. The observer is the one who sets the forward flow of time for all particles. Hence, to come back to the question, if I, the observer, call a system a refrigerator, I have already selected a time direction for it, which makes original question itself ill-posed. In other words, the reason why the refrigerator's time and the spoilage of food and the influence of gravity all have the same time direction is because their time direction is assigned by the observer.
Now, the second law states that entropy in a closed system must increase over time. Hence, just as before this defines a time direction for the property of entropy. Now, somehow we must synchronize this time direction with that of all the other time directions, again via an observer. The magical thing, as the article you cite notes, is that for some reason the time direction selected by the second law is always the same as the time direction selected by observers in the physical universe. But, as the article also acknowledges, nobody really knows why. At least, it doesn't seem that this is a derived consequence by any well-accepted physical theory, but rather a postulated coincidence (like the equivalence of inertial and gravitational mass).