> Sex is defined by the gametes one produces

It can be, it can be defined other ways; biomedically for humans its not usually defined exclusively by gametes, since humans who do not produce gametes are usually assigned a sex (recently, in medical practice, assigning more than one, scoped to a particular purpose, and potentially involving narrative rather than simple labels has had some adoption and more discussion, because “sex” broadly is a multidimensional space of correlated, but not perfectly correlated, traits.)

> Gender is the social expression of the set of behaviors typical for a given sex

That’s specifically gender expression, which as distinct from gender identity and ascribed gender as it is from any of definitions of ”sex”.