What "you've been told" could not be farther from the truth. During the fall, grass plants will build carbohydrate reserves. Over-grazing of those plants will not allow this to happen, thus your spring greenup/growth will be MUCH slower.
The only way to build up carbohydrate reserves in the grass plant is to let it grow in the fall (probably specfically the month of October). "It takes grass to grow grass" is an absolute truth. You need the plant growing to have enough leaf area to allow for proper /adequate photosynthesis which in turn allows for more growth which in turn allows the plant to build root reserves which allows it to survive the winter well. Your spring growth response will be MUCH greater.
To answer your question, you probably need a minimum of 4-8 inches of leaf area "left" before you start to graze those pastures in the fall. It does vary by species of grasses. When all top growth has ceased and the soil temperature drops to 40 degress, then you can successfully graze those pastures throughout the winter without any detrimental effects.