The modifier is for the E/W (aka latitude) factor since lines of latitude (aka, parallels) have to get smaller as you move poleward.
How accurate/representative/precise are your input values in the first place? (pick any of those terms, I doubt that your input data are 'continuous' to begin with but measured at discrete locations and interpolated. aka... precipitation, is never a 'continous' surface, it is an 'interpolated' surface but represented as continuous-ish.
As for the need to project in the first place, you indicate that you are needing 'zonal statistics'
So I would be as concerned about the geometry representation of your zones. Projected or unprojected, your 'zones' are only as good as the geometry used to make them.
I think "cell center" is the determining factor as to whether a cell is 'in' or 'out' of a zone, so the bigger the cell, the greater the potential for over or under representation … the latter is hard to get a handle on unless you experiment with zones of various shapes and configurations. For a example a long skinny zone may not get 'hit' by an overlain raster with a large cell size.
You could experiment... halve your cell size, increases the spatial resolution by 4X... not your accuracy, but the resolution, the hit/miss of a cell center and whether a cell will fall within one zone or another.
Lots to consider, but nothing will make a more "accurate map/analysis" just one that is "maybe ess wrong". Just qualify your conditions and draw your discussion within those bounds of qualification.