The case in your OP is Mumsnet following this procedure for website operators if they are sent a notice of complaint about allegedly defamatory material posted on their site. It's not to do with the proposed new laws, this is the defamation act 2013.
A poster posted some information which the complainant alleged was defamatory.
The complainant presumably reported the post and MNHQ decided it did not break talk guidelines so did not delete.
The complainant sent a notice of complaint to MNHQ. This gave Mumsnet a way out of being liable for the content. The procedure is to contact the poster and ask if they want the content removed or if not, to ask for full legal name and address, DoB and email. If the poster doesn't respond or sends obviously false details, MNHQ must delete the material. The poster's details are not passed to the complainant without permission but they must be provided to the website operator if they want the content to stand.
(Website operators still have the option of defending the defamation claim themselves but they'd have to really really care to choose a lengthy legal battle over just hitting delete or passing liability to the poster.)
In this case, the poster responded saying they didn't want the content removed, provided some contact details but did not give permission for the details to be passed on.
The complainant then applied for a court order to make MNHQ pass over the details, which they did. They also deleted the content.
This doesn't imply anything about the content or whether it was defamatory. MNHQ have complied with a court order to provide some information, that's all. They said they'd be willing to contest the court order if the poster asked them to but they didn't respond in time.
One possible ground for contesting such an order would be if you didn't believe the information would be used for the stated purpose. For example if someone told a judge they needed a mumsnet poster's real life details so they could take them to court for defamation when actually they intended to add the details to a database of Bad Women and then share this with their friends and followers.
And all of this so far is just about getting the name of someone for the complainant to take to court (assuming good faith). The complainant would then have to show that they have suffered serious harm to their reputation.
I have read the deleted content in this case and, while I found it very informative, it didn't alter my opinion of the complainant's character at all.
Even if the complainant could show that they have suffered serious harm to their reputation there are a number of defences that can be used by either the poster, the website operator or both.
In this case, the defences of truth and public interest are both relevant.