I am so sorry that Children Services are still causing you so much frustration and distress regarding the status of your grandchild i.e. ‘looked after’ or ‘private arrangement.
In your post of 22 June 2015 you said that your grandchild was subject to s.20 plus, an interim supervision order. The fact that your grandchild is under s.20 means she is voluntarily accommodated by Children Services and therefore a looked after child.
Also, from what you have said about how your grandchild came to be in your care it is difficult to understand how this can be said to be a private arrangement. You said in earlier post that your grandchild was removed from a previous carer which appears to have been because Children Services was not happy with the care of your grandchild.
Have you asked the social worker to put in writing reasons why she considers that your grandchild is not a looked after child? I suggest that you ask for it to be put in writing so you have evidence should you make a complaint. Please also keep a copy of your letter and copy it to the head of Children Services. It might help if you mention in your letter this case R(D)-v- Southwark LBC  EWCA Civ 182 in this case the Court of Appeal said if Children Services played a major role in placing a child then it is more likely for child to be treated as looked after and not a private arrangement.
You are absolutely right not to be bullied by Children Services into accepting that your grandchild is not looked after as the level of their involvement suggests that she is and the case law is there to support your position. Please read pages 5 to 6 and 15 onwards of the advice sheet here regarding support Children Services should offer. I have also included a copy of our advice sheet about the duties Children Services have where a child is in the care system.
I suggest that you telephone our advice line on advice line on 0808 801 0366 should you wish to speak to an Adviser for more advice. The advice line is open Monday to Friday from 9.30 a.m. to 3.00 p.m.
I hope you will find this information helpful.