Transgender, in its conception, was a coalitional term designed as an umbrella for all sorts of people who transgressed against cisheteronormative gender roles. This included transsexual people, but it also included crossdressers, drag queens/kings, stone butches, fairies, dykes, aggressives, removeds, and a whole slew of other identities (many of which would, in our current terminology, be considered "cis").

It was only in the late nineties and into the early aughts that the term transgender started being viewed as synonymous with transsexual. This has led to a lot of interesting (though often inflammatory) shifts in the language used in queer communities. In the anglosphere, the language of institutionalized queer organizing gained prominence, and street-level identifiers fell by the wayside. There were lots of reasons for this: some identities were considered too niche, or too difficult to parse for cishetero audiences. For some, the terms that were symbols of self-realization in some communities were often considered slurs in others (and this is especially true of identifiers used by racialized and otherwise marginalized communities, as able-bodied, educated, wealthy white queer people became a focus for deciding which language was acceptable and which was "offensive").

With the prominence of the coalitional term "transgender," which offered an opportunity to bridge the gap between a lot of different marginalized groups under a cohesive banner, transsexual came into a specific sort of cross-fire. On the one hand, you had a new wave of self-identified transgender people making arguments that transsexual as a term was "binary" and "reinforcing gender norms," which you may recognize as a parallel to arguments that "bisexual" as a term "reinforces the binary." (This is also a bit of a rehashing of the old lesbian movement's arguments that androgyny is the "correct" way to do lesbian feminism, and that femininity "reinforces the patriarchy." Turns out political movements are often doomed to recycle the same tired and divisive rhetoric).

On the other hand, you had transsexual people who did struggle with accepting or understanding the larger coalitional movement, for a variety of reasons. For instance, there are transsexual people who were resistant to the idea that they could be "lumped in" with crossdressers, or queens, because (especially at the time) many people who were openly transsexual lived "straight" lives, and couldn't agree with the fact of their manhood or womanhood being conflated with queer sexual practices. There were transsexual people who considered themselves to have a medical issue unrelated to queer activism, or who desired to live lives of stealth. There were transsexual people who saw their very identity as transsexual get villainized by other queer activists as "reinforcing the binary," as though some identities could be inherently radical/more radical than others. There were transsexual people who were having their very specific transsexual needs sidelined under wider discussions of transgender activism and transgender rights.

These were all very real and interlaced conflicts of language, the type that will come up in any coalitional organizing, by the way. Coalitions are great for getting people swinging together, but they can easily end up replicating systems of hierarchy and invisibilize the differing needs of the members within that coalition (check out Viviane K. Namaste's Invisible Lives: The Erasure of Transsexual and Transgendered People and Julia Serano's Excluded: Making Feminist and Queer Movements More Inclusive).

This is all to say that there has been a very deep interplay of competing ideas of what it even means to be transsexual and transgender, that there is no consensus and that there can be no consensus because any consensus would at its heart replicate the very systems of assignment of identity and gender role that transgender activism erupted to combat. There is a very real effort by the bourgeois institutions of queer theory to create a containing and hegemonic ideal of queer identity that can be easily captured and consumed in the commodity market, and this has coloured the way that queer identity is understood and discussed at large. There is no "correct" term for anyone to use, and you simply cannot judge a person based on what words they use to relate to their personal experiences. Language is always in motion, and while often that motion is being directed by the institutions of power, those on the margins will always carve their own linguistic space, and it is incumbent on us to allow people the opportunity to self-describe.