Gender identity and/or reassignment
Gender identity and gender reassignment are not the same thing.
There is no legal definition of gender identity in UK law. Where it is defined, it is described as an internal feeling.
There is a definition of gender reassignment in the UK Equality Act (as stated by pp). It describes a process of transition. Whatever that means we might argue about, especially from a feminist viewpoint, but various judgements allow us to conclude that this is not understood to be a mere innermost feeling. There has to be a stated intent at the very least, if not an action taken towards transitioning.
So conflating an ideological belief with an action (undertaken, in progress or planned) is not correct. As this is a list of the protected characteristics, it should be noted that they cannot be designated or changed at will - it takes an Act of Parliament to do so. Until that happens, the list of mandatory (i.e. named in the Equality Act) protected characteristics must be the ones stated in the Act. It is legal to add additional characteristics (such as class or education or gender identity) but even then the protected characteristics under the Equality Act take precedence.
The Act protects transgender people.
Well, to be precise, the Act protects transsexual people. However, its protections can be applied to transgender people, just not today's trans umbrella in its entirety.
The Act no longer requires a person to be under medical supervision to be protected, so a person who does not undergo any medical procedures would be protected.
This is an inaccurate representation of the Equality Act. It never required a medical transition before a transsexual could be protected under gender reassignment. And although it is correct that the Act protects those who have no intention to medically transition, someone who merely states "I identify as" but has no intention to change any aspect of themselves would not be protected. A social transition or even a plan to socially transition would however be enough to engage this protected characteristic.
The focus on a medical transition is also misleading, because their text could be interpreted to mean that while you don't have to medically transition, you do have to socially transition before you are protected under the characteristic of gender reassignment. You don't. An employee for instance who informs their employer that they are planning to come out as trans and, for instance, wish to be addressed by a different name or wear the uniform mandated for employees of the opposite sex, would be protected under gender reassignment from that moment onwards.