In an axiomatic theory, relations between primitive notions are restricted by axioms.
[1] Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading.
[3] Alfred Tarski explained the role of primitive notions as follows:[4] An inevitable regress to primitive notions in the theory of knowledge was explained by Gilbert de B. Robinson: The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics: In his book on philosophy of mathematics, The Principles of Mathematics Bertrand Russell used the following notions: for class-calculus (set theory), he used relations, taking set membership as a primitive notion.
(p 25) As for denotation of objects by description, Russell acknowledges that a primitive notion is involved.
(p 27) The thesis of Russell’s book is "Pure mathematics uses only a few notions, and these are logical constants."