Number of families is irrelevant really as a business has to have (or intend to have) many clients over a period of time. I think a lot of confusion over the 3 families thing is to do with Childcare law where a childminder is someone who cares for the children from three or more families.
Mutuality Of Obligation is the main thing I would look at. Although no one factor is used to determine employment status, thus one is very important as it is about the relationship between the parties.
Someone who provides a service does so without the service user being obligated to ever use that service again. Think of it this way; you ask a plumber to fix a leak, they fix it, you may never see that plumber again.
In a childcare context a babysitter is running their own small business providing a service. Parents use the service when they need it and when it suits the babysitter to accept the booking. The parents may book that babysitter again in the future or they may not, they are not under any obligation to book that babysitter again and the babysitter is not under any obligation to work for that family again.
A nanny is therefore an employee as they are contracted to work specific days/times and are doing so on a very regular basis. It is more than having a babysitter who is rebooked repeatedly as both parties concerned view each other as being obligated to provide/do the work. The nanny cannot simply say they are not coming in, or don't want to keep doing the work. The parents cannot simply tell the nanny not to come in any more.
Running a business you would be doing a one off job for a client, it could be just one day, a couple of days (I am working this weekend for a family via my business), or maybe a week. If it became many weeks then it could become employment.
So look at each bit of work being done, is it occasional without any obligation to provide the service? If so then it may be you providing a client with a service.