I like crazyheart's statement "If you think you are family, then you are family."
I think of myself, my husband, our son, our daughter and grandchildren as family - and I include my daughter's significant other as part of the family as well. But I also have my family of origin - my parents (deceased), my brother (deceased), and my two sisters (but not their husbands and children). My husband has his family of origin - his brothers.
I also mentally define 'family' in context. When my granddaughter was little, she and my daughter lived with me. We continued to attend church together even when they moved out. I was very hurt when I signed us up to attend a church social event as 'family' and was told that we were not a family because 'we didn't live under one roof'.
I generally think of a couple as being a family. Or a couple with a parent living with them. Or two sisters sharing their home - or two brothers. Or a mother and her adult son - or daughter. And yes, I can well imagine somebody explaining, when asked about their family "There's just me and my cat, Fluffy."
But 'starting a family'? I would think of that as a person, or a couple, deciding to have (or adopt) a baby or child.