This is a huge challenge, and I suspect/hope the growth in analytics will soon make pixel-based motion detection obsolete.
The way pixel-based motion detection works, you typically have just two dials to turn. One adjusts how sensitive the pixels are. For example if you convert to grayscale you could say on the scale of 0-255, the pixels must change by at least 50.
Then you have the total number of pixels that must change before it is considered "motion". This is your "object size" except I think most of the time the total is measured on the whole image. If there is a pixel-based motion detection algorithm out there which allows you to specify that these changing pixels must be grouped together into an "object" with a defined minimum width/height and density, then that might actually defeat snow. I'm not sure if this exists, but if it does, it already rides the border between motion detection and analytics...
Since you mentioned Axis, you might look into Axis VMD 3 which is an add-on/plugin for Axis cameras to enable more intelligent edge-based motion detection.