How does one "count" filter order? This is primarily a question about how to name my filter.  Suppose that I have a photon detector, and that I want to measure the rate at which photons arrive, with a simple gated counter (because this is the only hardware I have).  The photon rate is just the counter value divided by the gate interval.

It seems to me that the counter is an integrator: so by opening the gate for, say, 1 second, I have made a first-order low pass filter (LPF).  

Now, suppose that I take the mean of, say, 10 such measurements.  Have I now built a second-order LPF? Or merely a first-order LPF with a longer time-constant?  Does the answer change if the measurements are not consecutive?
The easiest way to understand the concept of filter order is to look at the Laplace (continuous time) or Z-transform (discrete time) of the filter.  The filter order is the order of the transform's denominator.

For example, the Laplace transform of an integrator is $\frac{1}{s}$.  Extending the time of the integration does not change that it is an integrator, it just changes the bounds of the integration.  If you were to make it a double integration, though (two integrators in series), then the Laplace transform would be $\frac{1}{s^2}$ and you would have a 2nd order filter.

Averaging can be considered a partial integration with a gain factor, so your integrator with averaging would be a variant of a double integrator, making it a second order filter.