Gender is, according to the dictionary, the 'state of being male or female'. There is nothing there about feelings or choice, but there is a reference to social and cultural differences, as opposed to biological differences between the sexes.
Gender dysphoria is when someone's feeling about their gender is at odds with their biological reality.
Trans-people may or may not have gender dysphoria, depending on whether you believe in redefining the meaning of the word gender to mean an inner feeling about your gender identity.
Trans people cannot change their biological sex, but they can change the way they interact with people around them to conform more closely to the gender stereotypes of the opposite sex. If this makes them feel comfortable then it doesn't really matter with they have a medical condition called gender dysphoria or not, as it clearly helps them.
However, some trans people want to eradicate all reference to biological distinctions within our society, by competing against biological women rather than biological men in competitive sports. This is where women's rights should, in my opinion, trump trans rights. Similarly, places that protect vulnerable women cater for the unique vulnerabilities that come with being a biological woman, so those places need to be protected.
Trans activists who refuse to acknowledge that biology is a valid category upon which to make distinctions are not only scientifically wrong, they are also damaging women's rights and stripping our language of the vocabulary necessary to talk about women's issues.