My feelings on the transgender vs transsexual issue were greatly influenced by a trans man, the late Matt Kailey, whose books I read when I was first starting my transition. I describe myself as a transsexual male because I desire many of the physical characteristics traditionally associated with maleness, and have taken steps (hormones in my case) to get them. I normally just abbreviate transsexual to trans as I know the former term is so controversial and misunderstood.
But I am also agender as I don’t associate being a male/man or female/woman with any particular clothing, mannerisms, hobbies, or beliefs. So as a non-binary person, I fit under the “transgender umbrella” independent of my medical transition. Some non-binary people explicitly do not want to be grouped under that umbrella, however, and the same is true of many cross-dressers and drag performers.
As for intersex people, intersex trans male professor Cary Gabriel Costello has suggested using the term “ipso gender” rather than cisgender for those who agree with their birth designation. I would definitely not classify intersex people as transgender by default.
Regardless, if a person strongly identifies with the term transsexual but not transgender I can understand and respect that. I would probably still abbreviate to trans when referring to them unless they specifically requested otherwise though.