*I* would define homeless as someone who does not have a stable, consistant place to live.
Someone who has a room, or rooms (depending on how many people we're talking about) in a friends/relatives house and knows that space will continue to be available to them for an indefinite period of time (as long as they need it) and are actually able to live there (receive mail, have somewhere to put their stuff, etc) is not homeless. Someone who is staying on friends/relatives sofas, moving between various friends/relatives houses as they are welcome/sent packing - homeless.
Someone living on the street or in shelters - clearly homeless, no question there.
My guess would be that the school district goes with the "no consistant place to live = homeless" policy, basically. However, that the board doesn't have some sort of definition that is used to determine if a child is considered homeless or not seems very odd to me. Perhaps they need to discuss this with whoever runs the homeless education program - if those people don't know, then there isn't really a homeless education program (you can't serve a specific group if you don't know how you determine who the group is made up of)