There's an attempt to call „gender“ „socialinė lytis“ (= social sex). But doesn't make it sound like it's anything beyond socially constructed cultural phenomena.
So you...do have a word meaning "transgender?"
That has nothing to do with the word "transgender" existing in their language (which I would bet exists in pretty much any widely-used language these days, most of the time as a borrowed word).
Not sure of their language, but it is the same in russian, with the single word "пол" (pronounced as "pohl") used to refer to sex/gender.
The current usage of "gender" was introduced by American psychologist John Money in the latter half of the 20th century.
It's popularity in common speech was accelerated by "sex" becoming a synonym to "copulation," causing "gender" being used as a polite alternative.
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But how is gender actually defined? Even now, the Oxford Dictionary: "gender: the male sex or the female sex"