Then your central claim that sex is definitionally distinct from sexual determination is false, given that reproductive anatomy is sexually determined.
No.
Go read this. It's not super long, but it's very clear.
Defining vs Determining Sex
I think you're using a different meaning of "determined" than I am. Sex determination is the mechanism by which an individual is sent down a developmental pathway toward one sexual phenotype or the other. Sex definition is the phenotype.
Phenotypes vary by species, although they're largely similar among mammals. Our common evolutionary ancestor was already sexually reproductive when mammals branched off.
Determination mechanisms vary a lot among different sexually reproductive species. Among mammals, the determination mechanism is genetic, based on which genes are carried on which haploids, and how those two distinct haploids are carried by each sex. In humans, we can broadly say it's X vs Y chromosomes, but in reality, it's a specific gene - the SRY gene. This is normally carried on the Y chromosome, but there are genetic mutations that can disable that gene partially or completely, or can transpose it onto an X chromosome. That gene contains the instructional "prompt" that triggers an undifferentiated fetus to develop a male reproductive anatomy. If that gene isn't present, the fetus follows a Mullerian path and develops a female reproductive anatomy.
Things can - and do - go wrong with the instructions. Sometimes things get confused and bunged up.
But it still doesn't result in any individuals who are both male and female, or who are part male and part female. It can be very difficult to distinguish in some cases, but that doesn't mean they end up on some spectrum between male and female, creating a bimodal distribution of sex. It just means that you have to do more work before you can figure out which discrete box they belong in.
Sex is binary, it is not bimodal. Having individuals in a population that are hard to classify doesn't change that.