I think you could say that "drag" and "cross-dressing" and "transvestite" are synonyms, but there's a different shade of meaning. "Drag" is usually cross-dressing for the purpose of entertainment and caricature.

Merriam-Webster says:

"Drag" is entertainment in which performers caricature or challenge gender stereotypes (as by dressing in clothing that is stereotypical of another gender, by using exaggeratedly gendered mannerisms, or by combining elements of stereotypically male and female dress) and often wear elaborate or outrageous costumes
"… Atlanta … was, like, mecca for drag. It had the traditional drag queens who were female impersonators. But, you know, I had come from the punk rock side of the tracks, and we did drag as a social comment …"
—RuPaul

My honeymoon for my first marriage was a weekend in San Francisco in June 1971. My ex was a bit of a prude, so I was surprised that she agreed to go to the drag show at Finocchio's in the North Beach nightclub district. It was an excellent show, with great music and gorgeous costumes. As a good Catholic boy just a year out of the seminary, I didn't see anything objectionable about the show - and neither did my prudish ex-wife.