How does one "count" filter order?
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.