1. The problem with the definitions argument is that you were mostly using the diversity of definitions to throw up your hands and suggest that we couldn’t use them, bolstering the claim by picking out individual tricky cases. But that’s not a good argument because (a) a subset — maybe one — of the definitions might be stable and relevant, so you would have to argue against them all individually not just point out the diversity, and (b) the existence of tricky corner cases does not generally cause us to invalidate a definition but rather handle the corner cases, so rather than pointing out some cases where a definition would leave out people judged as in, you would have to argue that any sensible way of handling that corner case also admits trans people. (Perhaps a better example than the knife case is that “Please bring your own mug” is a perfectly understandable request, even if the exact boundaries of what a “mug” is is questionable, and there could reasonably be both structural and utilitarian definitions of a mug, but if someone shows up with a chopping board and says “this is my mug”, we would confidently say that no, they had not brought their own mug.)
2. Society has decided this, yes! Why? You could use this as an argument. Maybe it wouldn’t work, because you’re arguing against people who say that society should make different decisions, but at least it avoids the “society has to put me into whichever category I say, because I say so” problem that your previous argument had.
3. I’m not sure what your point is. Biologists have no trouble classifying organisms as male, female, both, mixed, neither, changing, whatever, depending on what the case is. My point is that you were making it sound like biologists have trouble with the concept of sex, or should, because it’s not a clearly binary phenotype. But it’s super-clear in all kinds of organisms almost all the time (including humans) because the vast majority have a set of commonalities with one sex and not the other, leading the population to naturally cluster into two groups. Any argument that doesn’t acknowledge this seems to be trying to avoid reality. If, having acknowledged this, we go on to point out that there are some cases that don’t fit neatly into the groups, or where the commonalities do not correlate in the normal way, that’s fine. But to the extent that you cast doubt on the underlying distribution of phenotypes, it just doesn’t reflect how biologists deal with biology. They note the commonalities, and then talk about the exceptions as exceptions — making separate categories for them if necessary. It’s not all shoehorned into “male” or “female”.

If you wanted to make the “I am female because I say I am” argument into a compelling one, you would have to do something like (a) argue that gender is relevant and sex is not, and (b) the principles of liberalism include great individual freedom in matters of central importance to people and that should include gender affiliation — it is precisely because gender is a social construct that liberal societies should welcome you to opt in. And we use “male, female, woman, man” etc. all to refer to gender as well as sex.

Otherwise, if you end up on the biology/medicine side, there’s actually no reason to be assigned a particular sex. Being “developmentally male and skeletally male, hormonally female, soft tissue sexual dimorphism mixed, epigenetic similarity score 0.54 male 0.35 female, etc. etc.” isn’t going to help decide whether you can apply for a scholarship or which restroom you’re expected to use. Medically, the binarization isn’t helpful either: knowing what cancer-prone tissues you have, etc., is the key.