From the link that you posted:
Sex refers to a set of biological attributes in humans and animals. It is primarily associated with physical and physiological features including chromosomes, gene expression, hormone levels and function, and reproductive/sexual anatomy
So then sex is an immutable trait. It is not "assigned" by anyone. It is identified at birth. There is no ambiguity.
Athletic competition is restricted to members of the same sex. Regardless of what your self-described gender is.
Sex is a biological factor based on genes and gene expression and it is not a binary. There are numerous genes that create primary and secondary sexual characteristics and numerous epigenetic factors that can affect the expression of those genes.
Gender is (seems?) biologically a result of brain physiology and does not necessarily match the biological sex. There is also potentially a factor based on society created definitions.
Athletics categories are explicitly decided on by people to fit their view of entertainment, profitability, and fairness. As of right now, the actual scientific data about advantages trans women has over cis women is ambiguous when the trans women have undergone hormone therapy and shows no advantages when the trans woman never went through male puberty.