First you should understand that being trans is far more than a physical change. It begins as an emotional need or desire. I have friends who have not altered anything about themselves physically for any number of reasons but live their lives as a female. Believe me they are no less trans than one who has gone "all the way".
The idea of legally changing genders is, in my view, a societal construct that has precisely zero to do with who or what I am. In fact I think that's a very complex issue. But it is a legal issue, not a definition of who we are.
You'll also find that bottom surgery, the creation of the opposite genders genitals, is a thing most of us do not have. Like the rest there are any number of reasons most of us do not get this done. Sometimes it is a question of money. With recovery and all the rest that is involved this can easily get to $100k in cost. Many of us, myself included, are not fortunate enough to have insurance that will cover it. My point being that bottom surgery is on no way a part of defining who we are.
You'll find much of society does want to force us into a nice, easily identified box. Trans must equal whatever definition they choose to apply. But the reality is far too complex for that. We are all different and not easily identified by any single definition.
My definition of trans is simple. I am Alyssa. I live my life as Alyssa. Out of necessity I do fill the role of the man I was before on rare occasions. But the masculine is just a role I play when needed. Some people who are falsely identified as trans are drag queens. Not all of them by any means but there are some who simply play the role of a female. After the show they wash off the glam makeup, put on their normal clothes and go home to their male life. It's a job. There's nothing wrong with what they do but some of them are no more trans than Tome Cruise is a fighter pilot or international spy.
I appreciate the fact that you are asking. I really wish more would. Hope I was helpful.