San1127, nobody said it was wrong for her to want to spend quality time with her son, it's even normal for a mother to want to do so but she doesn't want her Au Pair to be around them when she is off, ever.
To me it seems like she took an Au Pair for the price of it and just want someone to take care of her son when she can't and then hide in her room until her next shift. An Au Pair isn't supposed to be an employee, she is supposed to be treated as a member of her family, if OP wanted someone just to look after her son then she should have gone for a child-minder or alternative kind of childcare.
An Au Pair come for the cultural experience (hence the relatively low cost of an Au Pair) and it sickens me to see so many family take advantage of that and make them believe they'll be treated fairly to then turn them into a simple employee or modern slave.
you do not have to take an Au Pair into your home but if you do, make sure she is treated fairly and make her feel like home that's as simple as that tbh.
If OP doesn't want her Au Pair to join on outings she could just tell her '' X and I are going for a walk, we'll see you later, have a nice day ! " it's smooth and the Au Pair probably won't mind or be offended and if the AP ask to join she just have to say something along the line of '' X and I didn't spend much time together this week and would like to catch up and enjoy some 'mum and son' time together today, hope you don't mind, you are definitely invited to join us next time though."
There are always way to make you Au Pair understand that you would like some intimacy or time alone with your family, but you can't expect your Au Pair to never join you or hide in her room while the family is in the house.
I really don't this OP is suited to have an Au Pair and feel very bad for her AP, hope she finds a new family soon...