Example b) mother is in a relationship but not moved in with partner.
Example c) mother is in a relationship, partner helps her with money, spends time with kids.
Example d) mother is in a relationship. Lives with partner but still supports kids fully. (Money wise and gets a babysitter when needed)
Example e) mother is engaged but supporst kids fully.
Example f) mother is engaged but partner helps with bills paying at least half or majority of them.
So what best defines a single mother? Yes, you can have more than one example as the first one is an obvious yes.