A gender system (= a system of gender markers) is a system of non-°numerative °nomifiers which includes no more than 20 nomifiers.
Gender systems are often distinguished from other nominal classification systems by the notion of "agreement", but what exactly is agreement is not clear (see also Corbett 2008).
The definition proposed here is not based on "agreement", but on the notion of "nomifier", i.e. a classifier-like element in a nominal classification system (or nomification system; Haspelmath 2018).
The cut-off point of 20 is of course arbitrary, but if a system has many nomifiers, it is typically not regarded as a gender system (e.g. in Kilivila, an Austronesian language with over 150 nomifiers).
gender/class (SEM) = a semantic category that is expressed as an indexation feature, and covers both distinctions based primarily on sex and animacy (‘gender’) and distinctions based on animacy, sex, and various semantic properties of inanimate – or at least nonhuman – objects. (Section 4.4.2)
"Genders are classes of nouns reflected in the behavior of associated words." (Hockett 1958: 231)
"...the determining criterion of gender is agreement; this is the way in which the genders are "reflected in the behavior of associated words" in Hockett's [1958]definition given earlier. Saying that a language has three genders implies that there are three classes of nouns which can be distinguished syntactically by the agreements they take." (Corbett 1991: 4)