Groups like Stonewall have done a lot to muddy the waters, by using GI instead of GR.
In the Equality Act, gender reassignment means proposing to undergo, undergoing or having undergone a process to reassign your sex.
To be protected from gender reassignment discrimination, you do not need to have undergone any medical treatment or surgery to change from your birth sex to your preferred gender.
My understanding is that a person with the PC of gender reassignment cannot be treated less favourably than any other member of the population generally and any other member of their sex class when it comes to single sex spaces.
It does NOT mean that the PC of GR allows unfettered access to spaces reserved for the opposite sex.
But if an individual has a Gender Recognition Certificate, they should be treated as a member of their new (false) sex class, although can still be excluded in certain circumstances - changing rooms, toilets, hospital wards etc.
Gender Identity has no legal definition.
Actually I don't think Gender has, either, but I may be wrong there.