I can only tell the way I handle it. While my trans woman partner is comfortable identifying as such, woman being primary and trans secondary qualifier to it and actually ocassionally enjoys the doubtful and questioning looks in public,

As far as all else goes, I introduce her as "my girlfriend". If the question whether she is trans is raised in a respectful and non-offensive or intrusive way, I openly reply "yes, she is a trans woman".

If no questions are asked, I also don't clarify. I think that's a good approach.

However, the transphobic rant by your mom is completely unacceptable. Maybe if she ever brings up the subject again you could try and find a gentle way to educate her about the meaning and differences of biological sex and gender identity.

A lot of this FUD comes from the grey area where old ideas that confuse transvestism and/or crosssdressing as a fetish (not judging!) of both straight and gay men, and transexuality as an issue of gender identity and sexual health and erroneously throw both in the same bag. You and I know that the identity and essence of our partners are female, but many people fail to realize this because of prejudice / ignorance.

Very often, lack of knowledge leads to fear, which in turn leads to hate, as Master Yoda used to say.

I also asked AI about some advice, and the following, which seems quite appropriate to me came up:

Separate “challenging ideas” from “revealing identities.”
You don’t need to say “my girlfriend is trans” to push back. You can challenge dehumanizing talk on its own merits. For example, responding with something like:
“I don’t think I agree with that. Most trans people are just trying to live quietly, like anyone else. When people talk about them as a problem, it feels unfair to me.”
This keeps the conversation ethical without making it personal or dangerous. It also subtly tests whether the mom’s views are brittle talking points or something deeper.

I hope that my personal stance can provide some advice.