It sounds like you’ve got a good understanding, and like Cia and Nephylim confirmed, transpeople go by the gender to which they transitioned. If you’re not sure how to refer to someone, just ask “What pronouns do you prefer?” or something along those lines.
For myself, I think of the word transgender as a fact about myself more than as my gender. Think of it this way - if you were to meet a guy who was unattached, you would initially think of him as single. At some point, you may learn that he’s actually divorced. This doesn’t change his single status, it simply adds context to it. It’s a fact about him. Naturally, you might have a lot of follow up questions for him.
How long ago was he married? For how long?
Increasingly…was he married to a woman or a man?
What caused the divorce?
Does he have kids? If so, how is his relationship with them? Do they live with him?
Is he friendly with his ex?
Was the divorce agreeable or bitter?
How much of the story is shared is going to depend on your relationship with the person. Casual acquaintances aren’t likely to get the lowdown on all the details of what caused the divorce or the nuances of his relationship with his kids, and a casual acquaintance probably wouldn't even ask. But, that conversation should happen if a long-term, intimate relationship is on the table. Everyone’s story of divorce is going to be a little different.
Likewise with the trans population. Everyone’s story is going to be a little different, but the word itself is useful for giving a general fact about someone. There’s more to the story than just ‘male’ or ‘female’ just like with our guy above, there was more to the story than just ‘single’. Again, like above, it’s natural to be curious about the details but the level of information shared about the experience or the person’s thoughts about their trans experience is going to depend entirely on the relationship you have with them.