You just described winter in the Southern Tier of NY.Then yes we have winter! Foliage season is generally the last 2 weeks of September, first 2 of October. Somewhere in there it goes below freezing a few times then warms up for about a week: "Indian Summer". That is the kick in the pants that says "You better get things winterized, this is your ONE chance.
USUALLY we will see a little snow in late September and in October though it doesn't stick around. Didn't see any until December this year, super unusual. It is hanging around freezing currently, I think that is unusually warm for early January. Another USUALLY: we have a fair bit of cold then the "January thaw" where it goes above freezing for over 24 hours. That can be any time in January, early, mid, late. Following that we freeze our butts off as the temp dives well below 0°F. Plenty of single digits to negative temps well into February then it slowly warms up again. We always still see snow in March, often into April, on rare occasion in May. Spring really doesn't start until early May when the leaves on the trees finally come out.