Intersex people and the LGBT community.

Nowadays - maybe for the last year, so only very recently - I see more and more instances of articles and media, both gay and non gay related media sources using the acronym LGBTI. I meaning intersex. I'm just wondering how did that happen? Do intersex people consider themselves a part of the LGBT community?

I am asking because I have an intersex cousin and even though I have never talked to her about her being intersex (I really only meet her like once a year so we're not close) I am a 100% certain she would never think of herself as being a part of the LGBT community, in fact I am sure she is a casual homophobe just like the rest of my family and I have heard her use the term faggot just like everyone around me. The fact that she was born intersex was always treated like a secret shame in our family. My uncle drunkenly told the family that she was born intersex when she was already like 10 years old so her parents kept this even from the family like this is a shameful secret so this was very much treated as a problem and something shameful and to be kept hidden and my family members used to whisper about it like it was some horrible tragedy. I'm pretty sure she's never known any other intersex people in her life (or gay people for that matter, nobody is out where we live) so I know she wouldn't identify with the LGBT community at all. But she was raised as a girl and identifies as a girl and looks like a girl etc and is straight.

I'm wondering do other intersex people like getting mentioned alongside the gays and the trans? I imagine that is the case where maybe you were raised to be one sex but felt like the other etc... I guess I am a bit uneducated about intersex people! Can someone shed some light please?