Wikipedia has an article http://en.wikipedia.org/wiki/Non-Klinefelter_XXY. Off the top of my head, I know of no other sources, as the whole list was pulled from memory, and sexuality isn't my specialty when it comes to genetics.
"As best I can tell, we do have two (maybe three) sexes: one with testes, one with ovaries and perhaps one with ovatestes."
You choose to see it that way. Why you feel a need to simplify reality down to something more basic is beyond me, but it seems to be important to you. That need on your part, however, isn't reality.
A functioning SRY gene, for example, doesn't make you a male. It may cause - and does in all but a handful of edge cases - the development of male genitals, but it doesn't mean that the full suite of male hormones are extant and at levels that are normative.
Similarly, females can be androgenic which makes for a massively different life than non-androgenic females. It's a completely different sex, and it strongly contributes to gender identity, but isn't the whole of the basis for it.
Sex is about more than the phenotypic presentation of gonads. It involves everything from endocrine presentation (hormone levels) to neuroconstruction (which varies between the sexes, including neurotransmitter baselines.)
"Some people, for whatever reason, end up feeling they are the wrong phenotype. Those people would do anything to be be a "normal" man or woman."
Not all. Typically this presents in highly conservative cultures where prejudice is dominant. Elsewhere, where begin different is okay, the need to "be normal" isn't as important.
"Multiplying the number of sexes doesn't help."
Help what? Simplify things down to your binary view of sex? Reality doesn't tend to care how we choose to view the world.
"You're saying "hey these people might be in another category entirely"; it's more or less like a cis woman who says to a non-cis woman, "you don't have my experience, you have your own experience". "
The cis woman in your example would be correct.
"The argument can't be won by pointing at a gene or brain scan; the phenotype people feel is a subjective thing and that has to be accepted, before we even get to the performance of "gender"."
Wrong. Genetics and hormone levels can tell us biological gender. And there are a hell of a lot more than just two genders. Each biological gender will have it's own reality. The drive, experience, motivations, and biological impetus towards a gender identity will be different for each. For that matter, sexual preference is predominantly biological too, though some people do make a choice that is counter to their biological inputs. (There are, for example, plenty of documented cases where homosexual individuals chose heterosexual lifestyles in order to be normative.)
"Gender" is the result of all of this. Biological, environmental (epigenetic) and environmental (cultural) forcing all coming together to create the identity of the individual.