I would first define running in the snow as either aerobic running on snow covered roadways or maintained paths or trying to run through deeper unplowed snow.
I do run on compact snow and ice using regular running shoes or a pair that I've screwed TEK screws through the sole, this is a hardened tipped self-drillng metal screw, for really icy conditions. About ten per shoe is fine on a pair of shoes that are at the end of their cycle. I've never had any luck with any of the traction devices holding up to running.
I also run through deeper snow mostly with the dog and it is difficult work, you get anaerobic very quickly bustin through drifts, lifting your legs higher, postholing.
If you are going to run in deep snow invest in some racing snowshoes or cross country ski...if snow is deep you cannot see what's underneath and murphy's law is just waiting for you.
I do have a pair of adidas winter trainers but to be honest I usually just grab my shoes and the dog and go.
There also is a shoe called the icebug that has a kind of retractable stud in the sole and it's brother the dmg with a gaiter, you can read a review at runningtimes.com, shoe reviews. Frozen water is slippery, duh, so pay attention to foot plants and weight transfer,your stabilizer muscles are working a lot harder so tempo and mileage can't compare,try to breath through your nose or if it's colder through a face mask. Yellow glasses can help pick out dips and hollows, and you know some days I just walk, good luck.