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edit* there is the loop hole however that argues that if you initially leave your house to go to the shop (legal), but then while out, get a whatsapp from a mate asking if you want to come round for a beer and a burger, you technically are permitted by the law to do just that, as you didn't leave your house for that reason initially, and the act of leaving your house is the only thing that is specifically restricted.
Unless you have a child with you, in which case it would be illegal given the use of the words "outside the place where they are living" in regulation 8(5) as opposed to "leave the place where they are living" in regulation 6(1). Which makes no sense at all.
You're right, the direct restriction on gatherings is only in public places (Article 7),
However, if you have left your house to attend a bbq at someone elses house, you will have contravened elements of Article 6, so the gathering would still be illegal*.
https://www.legislation.gov.uk/uksi/2020/350/pdfs/uksi_20200350_en.pdf
edit* there is the loop hole however that argues that if you initially leave your house to go to the shop (legal), but then while out, get a whatsapp from a mate asking if you want to come round for a beer and a burger, you technically are permitted by the law to do just that, as you didn't leave your house for that reason initially, and the act of leaving your house is the only thing that is specifically restricted.
I'd suggest this is a perverse interpretation, but is an example of why poor legislative drafting can be a major headache down the line.