I confess that I have become very confused with this sort of thing.
In particular, Eddie Izzard used to describe himself as a 'transvestite' (specifically, an 'action transvestite - running, jumping, climbing trees, putting on makeup while you're there')
These days, he describes himself as 'transgender'. I'm not clear whether he now views himself differently, or if the vocabulary has changed and he has just updated with linguisitc shifts. I'm not aware that he's any kind of surgery or hormone treatments etc..
There are a number of terms that have gotten confused over the years. Most of the problem is simply that they are technical (medical/psychological) jargon, or "underground" code words from times when such things were less socially and legally acceptable, and the meanings have become distorted when said terms were discovered, picked up, and used more popularly. Sometimes deliberately so. Nothing unusual there, happens all the time, just look at the constant misuse of "quantum".
I'll explain a few of these terms; but keep in mind that these definitions are from a US-centric viewpoint, and other regions may have slightly different definitions or connotations.
A "crossdresser" is a cis-gendered person who dresses up in attire stereotypically attributed to the "opposite sex", predominantly for reasons of fashion. Cis men who dress up in clothing culturally assigned to women, and cis women who dress up in clothing culturally assigned to men. This was essentially an "underground" term, and has been used more broadly to also refer to transgendered/transexual persons, transvestites and drag queens/kings. Sexual preference varies widely, homosexuality is common, particularly among female crossdressers.
A "transvestite" is a cis-gendered person who dresses up in clothing culturally assigned to the opposite sex for the purposes of sexual gratification. It is a paraphilia, and was originally medical/psychological jargon that was adapted into popular use. The overwhelming majority of transvestites are male, most likely due to the fact that females wearing stereotypically male attire have not been as highly stigmatized historically. The term appears to have developed a somewhat different meaning in the UK idiom, used more similarly to "crossdresser", or to refer to transgender/transexual persons.
A "drag queen" is a cis-gendered homosexual male who dresses in an exaggeratedly stereotyped attire culturally assigned to the opposite sex, for the purpose of entertainment. A "drag king" is a similarly a cis-gendered homosexual woman dressing as an exaggeratedly stereotypical male. Drag queens/king is also sometimes used to refer to heterosexual performers as well, but some prefer to refer to them as "drag performers" instead, reserving "queen/king" for homosexual performers ("queen" is also a slang term for a gay man).
A "transsexual" is a person whose internal body image does not match their biological birth sex, and who typically has or intends to have (not many can, given the expense) hormone treatment and corrective surgery to transition to the sex that they identify with. This is, again, a medical term of art, but one that took on a slightly different meaning when adapted for popular use. At one point, the terms "transexual" and "trangender" were used to differentiate between those non-cis-identified persons who intended to physically transition to their identified gender, and those who did not intend to do so. The term has since been de-emphasized in favour of simply using "transgender" for non-cis-identified person, regardless of transition status.
"Transgender" refers to any person whose internal body image does not match their biological sex at birth. Originally it meant only those who identified as the "opposite" biological sex; but the term has since been expanded to include those whose internal body image does not strictly match either their birth biological sex, or the "opposite" biological sex, aka "non-binary" transgenderism. The term has also been more recently expanded to accommodate differences in cultural expressions of gender as well as internal body image.
