Thanks for quoting the Act, that is helpful. My understanding is that your interpretation is what it was intended to do: "protects people while they are undergoing gender reassignment via hormones or surgery".
However-
As you can see from a careful read of the excerpt you posted, there is no requirement actually specified in the Act for either hormones or surgery.
To have the protected characteristic of gender reassignment, the person has to be: proposing/undergoing/have already undergone
A process ('helpfully' not defined, so this could be anything)
By changing physiological or other attributes of sex.
Other could mean clothing. Or anything. Its not defined.
This together has the de facto result of anyone who is even thinking about transistioning has the protected characteristic of gender reassignment.
Its a terrible definition, but it does effectively allow for gender to be a PC. It would be almost impossible to come up for a legal argument as to why someone who believes they have the PC of gender reassignment, does not in fact have it. And its only relevant where someone is being illegally discriminated against - if you could prove illegal discrimination, you could probably also prove you have the PC of gender reassignment.
But the key part that the TRAs seem to miss is just because someone has the PC of gender reassignment, that doesn't allow them to claim to be the opposite sex.
Keeping TW out of women's single sex changing rooms does not equal illegal discrimination on the grounds of gender reassignment. Because their sex is male.
Re changing sex - strictly speaking the Act says gender reassignment is "for the purpose of reassigning the person's sex" - not 'changing' it. 'Reassigning' suggests relabelling or recategorising. Its what 'gender affirming' surgery used to be called.
Part of the problem with all these definitions is that they have changed over time and the wording of the Act itself is now 16 years old and showing its age - which is why there are all the references to 'transsexuals' - a term which has since fallen out of favour.
IANAL though!