Old school thought was that turf grasses should be cut short for the winter but the only reasons I could ever find for doing that were 1) to help keep leaves from being trapped by the longer grass blades, and 2) to keep the grass blades from being bent over by the snow. Even back in the past when I cut mine down, because that was then the recommendation I saw the "tunnels" voles made as they scurried around under the snow, so it does not make any difference to them how long your grass blades are.
In our area of the world the cool season grasses continue growing until covered by snow, or the temperatures go way down and stay there for some time, so cutting shorter will limit the amount of surface area the grass blades have to take in sunlight to make nutrients that the roots can store for use in the spring when the grass begins growing really well again. There is no benefit to cutting the grass shorter and only detriments.