Yeah it's the first time I've heard of it referred to as a spectrum. Although the term is insensitive it's more like two sides with a few outliers.

You have your XX females and your XY males making up 99.9% of the population and then some individuals who are XXY, XYY, or XXX. You can even have some who are XXXY, XXYY, or XXXX.

In terms of how that affects biological sexual development and associated gender identity i can't say offhand and it would likely be a rabbit hole that one can spend hours looking into.

https://youtu.be/kT0HJkr1jj4?si=_gOsunaQoUZ0bg0W

TL;DW: your understanding is a simplification. It an assumption that allows educators to move past complex nuances when teaching about broader topics as part of a general education. Chromosomes and gene expression are a chaotic mess in reality.

It's not really about the genotype. Every context that uses sex to divide people into groups has its own definition of sex. Sometimes with a spectrum sometimes not.

Example car crash testing: When testing safety features of cars, you're not gonna care about whether the person has a vagina or not, but features like height, neck strength and other stuff will matter.
Since women are on average smaller and less muscular than men, small people with weak necks will be categorized as female. Tall people with strong necks will be categorized as male.

Example Condoms:
Two categories: has penis (male, can use) and has no penis (not male, cannot use [for inteded purpose]). So if you are XY genotype, but without a penis (idk cut off?), you're not a male in that context.

As you see both examples correlate with the gonosomes, but are not defined by it.

Some figures claim that 1.7 percent of the population is some form of intersex, which is more common than having red hair I believe.