True, True - It's Confusing
What I think you should be looking for is an Intermediate SUV or Mini Van. Both of these would meet your needs without getting into the outrageous rates that would come with a Full Sized or Premium SUV. Car hire web sites won't be terribly specific about which actual make and model you will get, because they literally don't know which cars in their fleets will be there the day you arrive. Use a search site such as Travelocity or Orbitz to see what all the companies at a given location have to offer. Remember that Intermediate is the same as Mid-Sized etc.; they're just marketing terms. What you really want to look at is where each company will say "Ford Escape or similar" and then look up more details on the various models to get a feel for what you'll be getting.
AZBuck