Exactly Anya, a cross breed is a cross between two different breeds. Mongrels are dogs where there isn't a family tree to clearly identify which breeds donated to the current pups ancestry.
As the owner and foster person of doodle dogs, I'm well used to the breed snobbery they can incite in people who have no hesitation about sneering - oh an expensive mongrel. Nope, its either a rescue/foster or much loved family pet.
It's the breeding isn't it. So many pedigrees, like pugs, cavaliers, french bulldogs, german shepherds to name a few, bred to so called breed standard resulting in health problems. If you are considering a poodle cross, be sure to go to a breeder who is registered, isn't one of those places where several different types of dog are bred in huge battery type houses. A good breeder will give you a family tree, naming the dogs ancestors, will have health tested the parents and give you a certificate saying so. They will have handled the puppies regularly, exposed them to the kind of noises pups will hear in a family home, tv, radio, washing machine, hoover etc. Never buy from the internet. Never have a puppy delivered, you want to see where the pup has been living and meet the breeder. If you rescue, go through one of the specific breed rescues because their dogs will have been with experienced foster families so you will get a true idea of the dog you're adopting.