[okaram]

You are defining male and female that way. If you go by that definition, you can define a transgender person as someone who deeply thinks and feels they ought to be of the other gender. They are not 'mistaken' other than by your definition. If you ask them 'do you have a penis' or similar question, they will respond with your objective reality.

[jononomo]

Yes, I believe transgender people exist if you define the condition as people who wish they were of the opposite sex or mistakenly think they are of the opposite sex. They definitely really have mental conditions that result in wishes and feelings related to sex and gender, but if they claim that they truly are of the opposite sex, they are mistaken.