The linked question (with its answers) deals more with the formal aspects of definitions.
I'll try with a more "easy" approach.
Consider again set theory [ref. to Herbert Enderton, Elements of set theory (Academic Press, 1977)].
We will start from an intuitive explanation of what the topic of the theory is :
A set is a collection of things (called its members or elements), the collection being regarded as a single object. We write "$t \in A$" to say that $t$ is a member of $A$ [page 1].
This is not a definition : we cannot define everything and we have to start somewhere. This "elucidation" gives us the basic stuff of the piece of "Mathematical world" the theory will speak of : (some) objects and a (binary) relation between them.
Then a first "principle" is stated [page 2] : the Priciple of Extensionality. It states the fundamental property of sets : they are identified only through their members, i.e. only the membership relation is relevant for the world of sets.
This principle is used to show that the empty set (the set with no members at all) is unique, i.e. tehre are no two different sets that are both empty.
This little proof assumes that in the world of sets there is a set that is empty.
The next move is use the Axiomatic Method [page 10-on] to develop in a rigorous way the theory of sets. The method is well-known in mathematics :
we are going to state the axioms of set theory, and we are going to show that our theorems are consequences of those axioms.
The first axiom of the theory is the (previously stated) Axiom of Extensionality [page 17].
Next we have two existence axioms: the Empty Set Axiom, followed by the Pairing Axiom.
Nothing in principle has changed from the previous intuitive approach : we presuppose a "universe of discourse" for our theory and we call sets its objects. The objects are (in some cases) conected by the relation of membership, and this relation is "extensional".
In the universe of sets there is a "distinguished" object called the empty set.
Finally, for every two sets : $a$ and $b$, the universe of sets has also a new set (called its pair) whose elements are exactly $a$ and $b$.
The set existence axioms can now be used to justify the definition of symbols [previously] used informally. First of all, we want to define the symbol "$\emptyset$" [page 18].
The approach is quite clear : we have some assumptions (the universe of sets and the membership relation) that are so basic that we cannot state/define in the theory itself and with them we formulate axioms : some of them express basic properties of sets (Extensionalty) while other state the existence of specific sets.
When we have assumed the existence of a specific set (and showed its uniqueness) we may introduce a new symbol to denote it (a "name" for it).
Here the difference between axiom and definition is subtle : we have an axiom asserting the existence of a set with no memebers, and we define a name for it : empty set.
Conclusion : we have not defined what "set" and "membership" are. In the context of the specific theory we are developing, our knowledge of them is through the axioms.
We have stated axioms expressing properties of sets and membership and asserting the existence of sepecific sets.
We have introduced new names for those specific sets.