I think the existence of two separate words for the same thing may have caused a lot of trouble in the anglophone world. And..
That Butler should be ignored.
That gender is a synonym for sex (unless referring to grammar or copulation).
That it's most often used in compound nouns - gender norms, acquired gender, gender reassignment, gender recognition, gender identity.
That gender identity is a supposed sense of which sex category the self belongs in and therefore wishes to associate with: it does not entail 'feeling like' members of that category.
That trans people believe that they have a gender identity incongruent with their body, and this is proof that they are really the opposite sex: hence the emphasis on early medicalisation and rewriting the past. This belief is unfalsifiable.
That gender norms, roles, and stereotypes are irrelevant to the trans issue. They are just a collection of ideas about masculinity and femininity that are partly broadly natural and partly culturally contingent, but should not be forced on any one individual. They are useful to trans people for 'acting the part' of their 'true' sex.