is this definitely true?
Well, that image is not the wording the EHRC use. I think this is the really important change from the June 30th statement:
This means that a trans woman who does not hold a GRC and is therefore legally male would be treated as male for the purposes of the sex discrimination provisions, and a trans woman with a GRC would be treated as female. The sex discrimination exceptions in the Equality Act therefore apply differently to a trans person with a GRC or without a GRC.
There is certainly a strong argument to defend single sex provision that could be put forward by organisations like Girl Guides, but the problem is that it isn't clear cut.
This change from the EHRC is certainly very welcome, but I don't think the law itself has ever been what caused organisations like Topshop and GG to remove single sex provision. It's the threat of bad publicity, harassment and expensive legal action. These tactics are effective because the law is unclear.
I should add that IANAL, so this is just my musings on the subject.
The RHRC guidance is here:
www.equalityhumanrights.com/sites/default/files/what-equality-law-means-for-your-business-2018.pdf
Slightly more specific government guidance is here:
assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/85052/vcs-gender-reassignment.pdf