Transgender is a wide umbrella. All it really means is that your sex and your inward felt gender aren’t in sync. That need not be severe enough to cause dysphoria or even what’s called being gender non-conforming. Human’s love labels.
There are lots of kinds of trans people. Some do nothing about it, they’re happy enough, or they do something really small like a man getting his ears pierced or a woman wearing mannish clothes (and they might do it for other reasons too).
When someone has dysphoria then you start moving toward more major changes. Even then they may not transition. They might live part time, say in duality where they are Sal at work and Sally on the weekend. That might be enough for them, or for awhile.
Then some need to transition and live all the time, but they might need some or more or all of the interventions that can be done. I’m not positive but I believe the WPATH standards of care require dysphoria for HRT and SRS.
This inexactness of the definition is, I believe, part of the problem with how people view trans people with the bathroom/locker room issues. Because there aren’t different words to describe the different groups they have no way to understand what parts of that umbrella is really going to share spaces with them. Transsexual isn’t the right word though.
Imagine being a regular Joe or Jane. All you’ve ever heard about transgender people is indirectly and you’re trying to understand this. You do some small amount of looking online and you find out nobody really agrees on exactly what the word means. Are cross dressers transgender? What about those folks who just need to wear earrings, they’re transgender, right?
Then they hear transgender people want to share the bathroom with those they identify with and they are picturing this guy with an earring coming in, or someone in drag, not a trans woman who’s been on HRT. We need a word to separate the two.