The fuzzy zip-up should be good for a stroller (remember not to use any winter jackets or fluffy clothing in Carseats as they won't keep the baby in the Carseat in the event of an accident). I loved to borrow my husband's jackets and would zip them up around me and baby whenever I had him on me in the carrier. I would definitely recommend a snow suit for playing in the snow (you can usually get them pretty cheap at once upon a child). Snow suits will keep them dry and warm in the snow.
I moved from the desert to the cold so I get you! Usually a beanie, thermal top/bottoms, warm socks, and a jacket do the trick. If they’re in a stroller or carrier, I’ll use a couple of warm blankets on top of the warm clothes.