A gender identity cannot be defined. All that can be defined is what people who claim to have a gender identity seem to mean by it.
People who claim to have a gender identity seem to be claiming to have a gendered soul that exists separately to their sexed body. Or they are claiming that their brain's distress at their sexed body proves that their internal sense of gender is reality, not actual reality. Either way this soul or feeling is an incredibly important thing that defines them and proves that - simultaneously - they must be treated just like everyone else whilst being lauded as truly special because of their decision to announce their claim that they have a gender identity.
The implications of this seem to be that this gendered soul or gender dysphoric brain should matter more to society and everyone in it than the physical and unchangeable reality of the person's sexed body. This is the main problem with the insane TRA position. Even if it were true that everyone had a gendered soul that existed separately from their sexed body, or that gender dysphoric people are happiest if they are constantly affirmed, this does not mean that the vast majority of ordinary people should have their rights [that is to be segregated by sex in places such as toilets and sports and single-sex advocacy / support groups for all sorts of almost univeraslly accepted good reasons] removed.
A gender identity is basically very similar to a religious belief. Both are entirely relevant to the individual concerned and to other believers, whilst being absolutely meaningless to others. Everyone else has every right to ignore or mock the belief and assert truth - so long as they don't move beyond ignoring or mocking and start abusing or harrassing.
Explaining one's political views in your own home, in repose to someone else inappropriately bring up politics in the course of their job servicing the needs of customers, is not abuse or harrassment.