OP, I'd agree that there must be an additional concern from the HV for them to be dedicating so much time to your family. I do say this as someone who has both been taught safeguarding which was lead by a health visitor, and as someone who sees quite a lot of my sons HV, even with mitigating factors like quite severe MH issues, and a pretty fractured family, I see less of my HV than you do yours.
Can I ask, were you in care? Was your childhood abusive? What does the HV knowing your mums name suggest? What might her opinion be of you given the link between you and your mother? ....these things are how they make a lot of their judgements on the parents they see, and support they can offer.
TBH, I can see things that might be concerns, from what you've said, you're a young mum, who when you had your first child, didn't have family nearby, your HV says you're not engaged, you cancelled an appointment at late notice to go to soft play, you then refused her access to your children when she either didn't know the appt was cancelled, or was trying to do a welfare check.
In your position, I would call the HV team, and ask if you can discuss your concerns with her manager, and how you feel, and discuss from there whether you want to change HVs or if you want to completely ditch the service. Children's centres can offer many of the same services as a HV.
Your HV should be someone that makes you feel at ease and supported, not the way you're feeling.