The family rooms are the rooms where people often keep the TV, so they can save room for interactions that do not involve television. You will not have to worry about spills and accidents in your family room in relation to your more formal living room.
As long as you a comfortable place for your family to relax and have fun together, it does not matter where your family room. If you do not have the space on the first floor of your home, you can always have your family room in a finished basement or somewhere on the second floor.