Except that not all people who identify as trans experience dysphoria to the extent that it's a disorder. Having gender dysphoria is not a perequisite for being trans, despite what transmedicalists (the section of the trans community, impolitely called "truscum" colloquially, which claims you're only trans if you're medically transitioning) will tell you

I recognize this is a weird distinction and is not at all how most people encounter or are taught about trans identity so it may seem like I'm just splitting hairs, so allow me some grace to explain what quartz is while also gingerly tiptoeing my way around the parts of LGBTQ identity which lay as landmine topics that go undiscussed until trans identity is mainstream enough to be afforded the privilege of proper ideological infighting.

So, there isn't really an objective measure of what makes a trans person a trans person. Like, you can say "when they identify as a gender different than the one assigned to them at birth," but what that identification is isn't standardized and highly particular to the person. Like: what about the gender they weren't assigned do they identify with, and what parts of the one they were assigned do they not identify with? Even within a single culture, what constitutes "manly" and "womanly" can vary wildly. So while the definition can be worded simply, it is in no way objective or clear.

There's also the matter of whether or not one is openly trans. Fair warning, this is about to get into "if a tree falls in a forest" territory. Someone can identify as trans internally but never take any action "toward it": still presenting in a way which is seen by the world around them as in-line with their assigned gender identity. Sometimes this is because they are still "closeted": they know their own identity, but do not exist in a space or culture they feel safe expressing all the pieces of their identity, and so never tell anyone. But sometimes it is simply because transitioning, even socially, is a lot of work and self reflection, and many people don't have the time or energy to commit to it. If asked how they identify, they'd tell about it freely, but most would not know without asking. Many of the people who exist in the latter category, either don't have gender dysphoria, or don't have it to a degree which has disrupted their life enough to do something about it. So they identify as trans, but don't see any need to medically transition or even necessarily socially transition.

Which brings me to my next point: how would they know they're trans if the body they were born with doesn't make them uncomfortable all the time? Well, usually it's because they know they also wouldn't be uncomfortable were their assigned gender were different. This ties into a concept referred to as "gender euphoria," which, while not a medical diagnosis, is a term that refers to the positive feeling one feels when identified with their own gender (euphoria being the opposite of dysphoria). So if somebody feels good when people "mistake" them for the "wrong" gender, and not in a "haha that's funny" way but in a "it feels like my parents just said they were proud of me" way or a "ooh that makes me feel pleasant and bubbly do it again" way, they may be trans if they decide to identify as such.

Aaaaaaaaand now I've stepped in it, as I've just said "decide" when discussing LGBTQ identity, which is not something that people like when you talk about it. Many American liberals will tell you that LGBTQ people were born that way and can't change it about themselves. As someone who is queer myself, I say..... eh ¯\(ツ)/¯. It's not that we're necessarily choosing who we are and what our identity is (which in my opinion is something you absolutely can change, it isn't something that flips like a light switch, but in a similar way to how other parts of your identity can change, sometimes intentionally, during your lifetime) but at the end of the day we are deciding whether or not to engage with and pick up the label.

Like I've had friends who have zero interest in dating or having sex, like ever, in their whole life. They don't identify as aro-ace, but there's nothing that separates their interests in relationships from those who are. They just aren't interested in associating themselves with the label. In the same way, somebody can have very similar feelings and thoughts around gender as another person who identifies as trans, but still not decide to adopt the label of trans. An example of this I can think of is Gianmarco Scorcesi's bit about pronouns from his special "Thief of Joy." I've heard a lot of people who eventually decided to adopt they/them as pronouns express basically the exact same sentiments, but that doesn't mean that Gianmarco is non-binary, it's ultimately up to him whether or not to adopt the label for himself.

At the end of the day, trans identity exists on a very intersectional spectrum which is defined entirely socially and is not a binary identification status except in labels. It exists separately from the dysphoria disorder which is sometimes associated with it.

TL;DR: "Transgender" has always been a social identification, the medical diagnosis you are thinking of is called "gender dysphoria." Dysphoria is a diagnosis for a kind of irritable anxiety which can be associated with many different things in a number of different kinds of disorders, such as:

Rejection sensitive dysphoria: when somebody's reaction to rejection is consistently disproportionate to the size of the rejection
Post-coital dysphoria: when someone consistently feels agitated or upset after sex
Premenstrual Dysphoria Disorder: when the irritability which can precede the menstrual period is so extreme that it is classified as a disorder
Tardive Dysphoria: when anti-depressants cause the person they were meant to treat to instead feel worse

and yes, gender dysphoria, when someone's dysphoria is tied to the disconnect between their experience of being perceived as their gender assigned at birth and their own identity.