I am hardly an expert on this subject, but my understanding is that a transvestite refers only to the person's desire to wear the clothes of the opposite sex. I've never heard of a woman transvestite, though maybe they do exist. And the difference in clothes isn't so dramatic. A woman can easily get away with wearing a t-shirt, a sweatshirt, etc. and jeans. So what? It's not the same as a man wearing an evening gown and heels, jewelry and makeup. But the word "transvestite" is always about clothing.
The other two words you listed, Becky, I think have the same meaning. A transsexual (or transgender person) has come to believe they actually are the opposite gender. They feel "trapped" in the wrong body. In my opinion, this is a denial of reality and a kind of mental illness. I can't understand why the media are giving this tragic condition all this support, and acting like it's just another political movement, and their "rights" are being violated.
Also, I don't think it's at all the same thing as being gay. A gay person does not deny being the gender that he/she is.
I wonder, what happens if a transgender person is also gay? That's enough to make your head spin!