Near as I can tell, neither of these is actually a fallacy of equivocation.
When you first define a term (gender as sex, in the premise of your example), and then argue based on a different definition of the term you have just defined (gender as related to gender identity) without actually changing the topic to the different definition (as you use both in the same argument), you commit a fallacy of equivocation.
I will note here that you haven't actually stated that you take issue with my definition of gender identity, and as such my failure to acknowledge that you have made such a statement is, in fact, a fault of yours
No. Apr 4, '14 at 1:47pm and Apr 7, '14 at 10:42am were my responses upon the previous two occasions, which you failed to aknowledge. On both of these occasions, I addressed that disparity. I did not yet define gender identity, as I was not aware that that is what you were after, but I did indicate that they are different terms with different meanings.
As I cannot infer your definition of gender identity, please define it rather than stating that its meaning is self-evident.
At no point did I make such a statement, but whatever.
In my understanding, gender identity is the psychological representation of gender (id est, sex) with which an individual can most readily identify itself. In other words, the sex/gender that a person identifies as being, regardless of the sex/gender of the body.
[...] whereas the other three terms are unwieldy and difficult for them. And that conceptualization typically includes both the physical and social aspects of cisgender people.
Now we're getting somewhere. Why exactly is it more difficult, aside from what is mentioned in your fourth point?
The second reason follows from the first; because of either a lack of understanding or because of a lack of acceptance, people will use the term "gender" to refer to the social aspects of the above conceptualization.
[...] I am of the belief that the assertion that gender and sex are distinct is not born from trans people, but rather has been adopted and refined in reaction to the way that other people's usage of the term invalidates our existence.
These, as well as your closing statement, seem more like reasons not to use it in place of the other terms. If people so frequently confuse and misinterpret the intended meaning, it seems an odd choice of word for your purposes. If you were to coin a new term, rather than redefine an existing one, you would not run this risk (although, I admit, you may have a hard time having it generally recognized).
Finally, "gender" is a single word, whereas the other terms consist of multiple word phrases. And it is useful to have one term which groups the other three; while they are not identical, they are typically related to one another.
I would think that their specificity would make them more useful than one term. If you just say "gender", it may be difficult to figure out whether you mean the role, the identity, or the expression, as we can see as early as page 1.
Which contradictions?
These two:
how they do and do not relate to sex, and how they are and are not opposites?
Unless you mean "the ways in which they do relate to sex and the ways in which they don't" for the first, in which case it is only one contradiction.
@Moegreche:
Is it correct to say that you think the distinction between gender and sex is unnecessary? If so, is there a consequence of making this distinction - besides the fact that it's unnecessary?
Yes. I consider it to be both unnecessary and unwise; unwise due to the fact that gender is still being used in both senses and still carries the original biological meaning. It is at times difficult to discern which of the four meanings (sex, gender identity, gender role, and gender expression)someone is applying to it, as is evident throughout most of this thread.
Is political correctness a bad motivation for drawing a particular distinction or bringing in new terminology?
In my opinion, yes, but that is more related to euphamism treadmilling than it is to this topic. My reason for bringing up political correctness was merely to address the fact that, although it influences the evolution of language and it appears to favour such a distinction in some contexts, it has not actually changed the meaning of the term at this point.