https://lawhive.co.uk/knowledge-hub/conveyancing/when-do-you-legally-own-a-property/
That's a bit misleading though because any solicitor worth their salt will have done something (I forget exactly what) that prevents anyone else changing the title until the registration has been submitted. Any solicitor acting for a mortgagee will be required to do this to avoid any possibility that the restriction might not be added to the title, I guess where there's no mortgage then it's probably unlikely that the purchaser will have stipulated this explicitly but I expect it would still happen.
So it's certainly true that once your name appears on the title it's definitely yours (and why everybody should register to get updates when the title of a property is accessed because if someone manages to get their name onto the title then you could lose your home even if you're still living in it - and you'll have to pursue HMLR for compensation) but I'm not convinced that it's correct that legal title doesn't transfer when contracts are exchanged.
And if legal title doesn't transfer until registration is complete at HMLR then it's potentially a bit of a nightmare for tax purposes for jointly owned rental property and would require any jointly owned property to be registered at HMRC until the HMLR registration was completed as it wouldn't be a schedule 3A excluded trust.
I'm not exactly sure whether that site you linked is saying that legal title transfers when the registration is submitted to HMLR (which is the key point that the hold on updates to the title is designed to ensure) or when the registration finally is visible at HMLR (which can be a very long time afterwards)