RE: do deer like tall grass or mowed grass better?
Actually deer don't eat eat "grass" very often. When you see a whitetail in a field it is usually browsing on forbs, growing up between the grass shoots, which are native herbaceous tender or soft perennials. Most grasses, unless just starting to grow have a very high lignin (fiber) content and offer very low protein contents.
Often when you do see deer eating grass it is actually a wheat or oat that looks a lot like grass.
As the grass maturesand gets taller it will become very fibrous. if you take a clump of taller mature grass try to rip it. You will see how tough it is. Look at it closely and you will see the fibers sticking out where you tore it. Then take tender new shoots from a newly seeded lawn. It tears very easy and doesn't show those tell tale fibrous strands. deer need about 16% protein in their diets. Corn provides 6%. Grass provides about 2% crude protein. These animals are intelligent or conditioned enough to know what is good for them. They will walk past a mile of red oaks to get to some white oaks. They won't waste their time on most of your common lawn grasses.
Next time you see one feeding, mark the location and walk out there and see what has been nipped off or torn from browsing. I'll bet you that you will find forbs growing up between the garss nipped right off.