You seem confused. Heterosexual, homosexual, and lesbian are not gender identities. They are descriptions of sexual orientation -- i.e., descriptions of how people experience sexual attraction. By the way, many English-speaking LGBTQ people avoid using the term homosexual due to its toxic history as a diagnosis of a psychiatric disorder. We prefer "gay" to describe people who primarily or exclusively experience attraction to people of the same gender.
(In the French and Spanish speaking worlds, from my experience, "homosexual" is often considered more polite than "gay," though this varies by region and dialect.)
Gender identity refers to experiencing gender differently from the sex one was assigned at birth. "Lesbian" is not a gender identity, but "transgender man" is. A transgender man is a person assigned female at birth, due to external genitalia and so forth, but who identifies and lives (to one degree or another, it's complicated) as a man.
Of course, cisgender (meaning not-transgender) people also have gender identities. Those identities just happen to match gender assigned at birth.
Cisgender people like me can be gay, straight, bisexual, etc. So can transgender people. Some trans folks identify as gay, others as straight, and some as bisexual. Pansexual is an emerging label for sexual orientation that some people use to make clear they are attracted to people regardless of gender.
Nonbinary is also a gender identity. It can refer to people who see themselves as living outside of the gender binary, but it can also refer to gender fluidity, a topic I'm not well qualified to discuss.
I think the take-home for me is that members of gender and sexual minorities are often treated as The Other (stigmatized as you say) because of our differences, which are quite real whether we emphasize them or not.
Take Transportation Secretary Pete Buttigieg for example. He's the constant target of homophobic slurs from the right despite being just about as heteronormative as a queer person can be. Meaning he looks and acts like a typical straight, cisgender man -- right down to monogamous marriage, two children, conservative dress, mannerisms, speech, etc.
He certainly isn't emphasizing that he's different from most people, even though he is. He plays those differences down to the best of his ability. But just this past weekend, Fox News commentators skewered him (again) in raw homophobic terms for taking time off work last year to care for his newborns.
Emphasizing his similarities has not helped people get along with him. And he's just one example.
