Benjamin wrote at a different time when "transsexual" was a relatively new concept. He was a pupil of Magnus Hirschfeld who had first described what we would call the "transgender spectrum", extensively and without judgment. He was the first one who did not pathologise trans people (in opposition to Freud and others), and he made clear that trans people are neither insane nor perverted.
In accordance with the time (~1900) he used the term transvestism (-vestism=cothes), because back then physical transition wasn't available yet (though there had been ftm surgeries such as mastectomies since the late 19th century). Clothes were the main tool that trans people used to transform their bodies. Hirschfeld explained that clothes become an extension of the body and can help to not only change the look but the internal body sensation.
He also promoted the so-called "third sex/gender" theory, but what he actually taught was that sexuality, gender expression, gender identity and physical sex are a spectrum with infinite individual combinations.
He later coined the term "extreme transvestites" for trans people who were looking for ways to change the body (such as self surgery). This later became "transsexual". But he also made it clear that transvestism and transsexuality are not separate phenomena, and the motivation is mostly the feeling that one is to some smaller or larger degree trans.
Things got messed up when Freud entered the scene and promoted the idea that transvestism is a sexually motivated perversion or fetishism. You could say that Freud stole the term and the whole concept of transvestism from Hirschfeld, and twisted it.
Benjamin also didn't say that there are "true" transsexuals as opposed to "not true" ones. The scale mostly had the practical purpose to help people (self) diagnose how important physical and social transition was for them, or if they could survive without it. At the time physical transition was still a high risk experimental procedure, so not putting everybody through surgery was an ethical thing to do. Several people died after surgeries or there were after effects such as incontinence or paralysis of the lower body. Hormone dosages were also still very experimental.
Both Benjamin and Hirschfeld believed in a physical, social and mental spectrum of sex/gender/sexuality, and today's concept of "true brain sex transsexuality" would have been alien to them, because they didn't believe that most cis people were 100% physically and mentally female or male, either.
The need for transsexuals to call themselves "true" transsexuals historically arose from the need to distance themselves from the Freudian fetishistic/perverted cross dresser concept that became popular since about the 1950s.
