https://en.wikipedia.org/wiki/Intersex
As for diagnostic criteria, they vary from country to country. Here's whats in the DSM 5
https://www.psychiatry.org/patients-families/gender-dysphori...
The second link basically confirms what I suspected.
I don't see how your appeal to tradition is sufficient to justify your view.
Science generally accepts that the identities it discusses are nominalist and conditional. We humans create boundaries and classifications by grouping elements, but no such group or organization is objectively correct, some are just subjectively useful for particular purposes.
I'm not appealing to tradition, I'm arguing for it based on what I've understood about gender dysphoria or the concept of gender in general so far. To reiterate, gender dysphoria comes with all sorts of potential problems, like medication, surgeries, depression, discrimination etc. And that makes me believe that the best way to go about it for parents is to make sure (to a reasonable degree) that the child will not develop gender dysphoria and that the biological sex of the child will match the gender (a social construct, so we can influence it by controlling the environment). Unless you want to argue that what we've been doing up to this point actually makes gender dysphoria more common and the 'new way' suppresses it, or that I'm missing something important, I don't see any gaps in my reasoning.
My question is specifically how do we diagnose gender dysphoria. Is it just based on the fact that a person might express himself in a way that society associated with one gender or another?
That would make sense to me. For what it's worth, from my observations of this community it appears to me that at the very least a significant part of self-{identified,diagnosed} transgender population seems to have an unrealistic, hypersexualized portrayal of what being a woman means, most likely influenced by pornography.
But why obscure and attack the concept of biological sex then? Doesn't that only reaffirm that what we've been doing this entire time is a right thing to do? Imposing the correct gender role (or as you called it, assigning gender at birth), seems exactly like the way to go, as it lowers the chance of child developing gender dysphoria and having all the problems associated with it.