Correct me in my following understanding of why I consider the following (in bold and italics) to be primitive undefined notions.
Variables stand for the mathematical objects that base the theory. Proper axioms tell us not what they “are”, but what we can “do” with them.
Functions and relations connecting the mathematical objects. Like addition and multiplication in linear algebra are described by the axioms. Similarly, belonging is also a primitive in set theory. Equality is also described by equality axioms.
One of existential or universal quantifier cuz the one can be defined in terms of the other via connectives.
For connectives I’m unsure. Because I think they can be “defined” using truth tables.