I may not be a professional scholar of human sexuality (although I am certainly an amateur academic), and it's worth disclaiming that, in spite of my unconventional experiences with sex and gender (or perhaps because of them, as even the sexual counterculture is becoming something of an establishment - it's sad, if not too surprising, but I feel that traditional LGBT discourse is too limiting). And I understand that language is fluid (although, again, this is a fact that could actually be used to support my position), and that people use words for specific reasons, even as those reasons vary from person to person.
But I am a scientist, and I like to organize concepts and look for neat, geometrical patterns. So, I may be trying to fit the infinite variations of reality into an artificially limited space (which is ironic, I guess, since I was criticizing exactly that only sentences ago), but this is how science works - by creating frameworks that help us to understand and make sense of the world (always with the possibility of being modified and improved in the future, as our understanding continues to evolve).
So, it is as someone who does not have first-hand experience of the historical perspectives surrounding the label "transgender", but someone who is trying to make better sense of the at-times convoluted and confusing language we use (especially given how sensitive some people can be when you misuse it - I've always been very forgiving of these kinds of metagrammatical errors, as they often stem from mere ignorance, as opposed to active hostility), that I propose the following usage of the various core terms related to cis- and trans- gender and sexuality. Even if it turns out to be not such a pragmatic construction, I still think it has value from a conceptual standpoint.
I'm starting with the basic understanding of the difference between sex and gender, with the idea that your sex is basically the biological body you're in - which includes your sexual anatomy. Roughly speaking, males have penises, and females have vaginas. And gender has more to do with your mind - your personality traits and identity, how you dress, how you behave, how you see yourself - not in the mirror, but in your mind. Traditionally, feminine gender cues are those that have been associated with females, and masculine gender cues are those that have been associated with males. Most people are born cis-gendered - that is, their gender aligns with their sex in the traditional manner (i.e., males are masculine, and females are feminine). However, there exist a minority of people whose gender does not match their biological sex - we call these trans-gendered. The prefix "cis-" indicates alignment, while "trans-" denotes a discrepancy, as regards the traditional formulation of gender.
Now, some trans-gendered individuals are content to live their life out as they are. I believe these could be considered non-op (this is where I, personally, stand). Others - and this is what constitutes the mainstream discourse on trans-sexuality - choose to embark on a medical journey in order to align their sex with the gender they identify as. Typically, as I understand it, they will undergo hormone therapy, and eventually have sexual reassignment surgery, in which their bodies will be modified to resemble the sex other than the one they were born as, so that they may live out the rest of their lives without disparity between their physical sex and mental gender. In other words, the goal of their journey is to become cis-gendered - to have a sex and gender that match. They weren't born this way, but that's where they've ended up.
Now, I feel that this is a bit of a change from how these terms have been conventionally used. I want to mark a difference between using the suffixes -sexual and -gender (or -gendered), the same way that we've distinguished the difference between a person's sex and gender. (Up 'til now, I feel that they've been used pretty much interchangeably - i.e., transgendered is a synonym for transsexual, and cisgendered is a synonym for cissexual). Remember that, in my conception, "cis-" indicates alignment, while "trans-" indicates a discrepancy. So, a cis-gendered individual is one whose gender matches their sex (according to traditional standards). On the other hand, a trans-gendered person is one whose gender does not match their sex. Cis-sexual, now, is going to mean anyone who remains the sex they were born as. Trans-sexuals, then, are those who have undergone (or, possibly, are currently undergoing) sexual reassignment surgery, and have changed sex. They are no longer the sex they were born as.
Cis-gendered - your gender matches your sex
Trans-gendered - your sex and gender do not match
Cis-sexual - you are the sex you were born as
Trans-sexual - you changed sex
So, in this conception, a trans-sexual is a trans-gendered individual who becomes cis-gendered by changing their sex. (Obviously, they don't have to broadcast that fact, once they've completed the change - that's their decision). To illustrate this visually, we can look at the sex/gender graph I produced for my SGO Notation. Let me derive it for you here.
We start with a typical mathematical graph on two axes. The horizontal axis is anatomical sex - biological males are above the axis, and biological females below. The vertical axis is gender - to the left is masculine gender traits, and to the right is feminine gender traits.
The statistical norm - cis-gendered individuals - are masculine males and feminine females, which exist in the upper left and bottom right quadrants of the graph, respectively.
The central area along the vertical (gender) axis, where masculine and feminine traits meet, is the androgynous zone.
We can perceive of individuals crossing the gender axis to exhibit gender cues that do not align with their sex. We call these trans-gendered individuals.
Trans-sexuality, then, is the phenomenon by which a trans-gendered individual changes their sex (crossing the sex axis). You can see by the way the sections are lined up, that when a trans-gendered individual changes their sex (but maintains the same gender), they become cis-gendered (their sex now matches their gender). They may still differ from non-transsexuals in that they have undergone a sex change - we can indicate that by labeling the default (those that have not changed their sex) cis-sexual. This graph does not really have a way to show that - we would, perhaps, need a third axis to indicate the passage of time. Cis-individuals would not do a lot of moving around, while trans-individuals would. We could also, then, depict bigendered and gender fluid individuals by showing their movement across the gender axis.
I like this conception, because it makes sense to me, and it creates much-needed symmetry between the way we talk about sex and gender, and using those same terms as suffixes in conjunction with "cis-" and "trans-". It also preserves the mainstream goal of trans-gendered individuals to align their sex and gender, in a sense becoming "normal". At the same time, however, I've come to this understanding because I want to make room for those trans-gendered individuals (like myself) who do not fit the stereotype - born into the wrong body, suffering body dysphoria - and, welcoming the transgender identity, do not wish to one day become aligned (essentially no longer being transgender - as if that were the problem to start with, and not society's expectations that the two sexes should behave in particular ways), but delight in flouting society's conventions and expectations, with an eye to encouraging an even more diverse and tolerant world, where people are free to express themselves in unconventional ways, without fear of unwarranted harassment or discrimination.