Is there a difference in laws or rights of a person that is living in someone's home,does not pay rent, has no signed lease agreement but pays utilities, groceries,and exchanges work for rent money,the same as a peson who signs a lease agreement and pays rent.Would this person be titled a lodger? What is the difference if any in California?

A lodger is a single (as in sole , not unmarried) person living in an owner occupied residence. Anyone else is a tenant. CA law allows for removal of a lodger without going to court. Otherwise the rights are the same.