The determination of sex in a biological context is based on chromosomes, hormone levels, and secondary sex characteristics.
For a 101 understanding of biology, by far the most common chromosome possibilities are XX and XY, but XXY is possible, as are mutations in one chromosome that combine pieces of X and Y such that a mix of secondary sex characteristics appear. It's also possible for someone with XX chromosomes to have abnormally high levels of testosterone, or someone with XY chromosomes to have abnormally low levels of testosterone and high levels of estrogen, with results such that a visual inspection would not be able to accurately guess the underlying chromosomes.
I agree with you regarding the chromosome possibilities. However, aside from rare mutations and developmental anomalies, mammals are either XX or XY. The exceptions prove the rule, and one doesn’t discard all that evidence just to accommodate minor anomalies which have adequate explanations. If you did, science would make no progress.
Regarding your first paragraph, replace “biological” with “sociological”. Then, that is what has been done traditionally, in the absence of the recent ability to determine sexual genotype by DNA analysis.
The Wikipedia pages on Sex[0] and Phenotypic trait[1] are actually pretty neutral.
Bimodal still implies that most things fall into one of two categories. Binary implies that they all do. You've basically admitted that the distinction is not binary.
Further, we still don't do DNA analysis on the vast majority of people to determine whether they are biologically male or female, certainly not before making a determination for a birth certificate. Even if we did, there's not one true X chromosome and one true Y chromosome that all XY people have, which a strictly binary interpretation would imply. Genetic crossovers between the two are a natural part of reproduction.
The biology of this is really fascinating, and I think it's both disingenuous and incurious to attempt to reduce all of this to a strictly binary model.
Male and female genotypes don't form a distribution, they are one or the other, modulo very rare transcription errors or mutations (which generally don't survive the first few cell divisions following fertilization).
So, talking about "bimodal" as if they were a combination of two normal distributions is not reflective of objective reality, since neither one is stochastic. There is no normal distribution of male genotype-ness, and no super male chromosome.
All X chromosomes differ from all other X chromosomes, but there are no half-X, half-Y chromosomes.
Because of this, in virtually all cases sex at birth is obvious to a medical practitioner. In questionable cases exhibiting abnormal genitalia we can, these days, do a quick DNA test to decide the matter. Wishful thinking is unnecessary.
For a 101 understanding of biology, by far the most common chromosome possibilities are XX and XY, but XXY is possible, as are mutations in one chromosome that combine pieces of X and Y such that a mix of secondary sex characteristics appear. It's also possible for someone with XX chromosomes to have abnormally high levels of testosterone, or someone with XY chromosomes to have abnormally low levels of testosterone and high levels of estrogen, with results such that a visual inspection would not be able to accurately guess the underlying chromosomes.
Hence, bimodal, not binary.