Well, file this under posts of the type I don't usually make anymore. But trans* language is a funny thing. It changes so fast that even trans* people don't agree on what it is. A couple of years ago, I was a guest speaker at the Transgender Education Association meeting. I was the education for the evening. But I arrived early and before the meeting started, one if the attendees, who had no clue who I was or why I was there, decided to explain to me what the words "transgender" and "transsexual" meant. According to this person, "a transgender" is a person who likes to dress in opposite sex clothing now and then whereas a transsexual is someone who desires to change their body.
GLAAD will tell you that "transgender" is an umbrella term that encompasses anyone whose gender identity and/or gender expression differs from their gender assigned at birth. Many, if not most dictionaries, will tell you that transgender refers to someone born one sex, but identifying as the opposite. Some will say transgender and trassexual mean exactly the same thing. Merriam-Webster has a definition on their website that seems to say only nonbinary people are transgender. All of this, of course, is a way of saying that language around trans* issues is messy, confusing, and not always universally understood by everyone in the same way.
I don't know that I've ever heard anyone say that intersex is a type of transgender identity. You do encounter quite a few intersex people in the trans* community which may not be surprising considering the number of intersex people who have ambiguous genitals that get "fixed" by being assigned to make or female when they are born. Sometimes the "fixer" gets it wrong.
There are people out there who identify as intersex, who also identify as trans* (I am one). But there are also a lot of intersex people who are fine with whatever gender they were assigned to at birth who don't consider themselves trans* at all.
We all can have many identities outside of trans*ness. For example, if you spell me out, then I would identify as an intersex biologically androgynous female identified asexual heterosexual leaning biromantic person. But the intersex is an identity I have that is separate from the trans*. It's an additional thing (that I don't much like to talk about). I would hope others will accept and embrace me as a trans* person . . . who is other things as well. One of those other things being intersex.