The first appearance of the term “model” as name for an interpretation that satisfies a set of formulas (or theory) is von Neumann's Eine Axiomatisierung der Mengenlehre (1925), §4. MODELS OF SET THEORY.
It seems that the new term owes its success to Hermann Weyl’s Philosophy of Mathematics and Natural Science (1st German ed., 1927), page 20 :
An axiom system must under all circumstances be free from contradictions, in which case it is called consistent [...]. Not indispensable but desirable is the independence of the individual axioms of an axiom system.
There are three methods at one's disposal of reaching this goal [to establish the independence of a proposition $a$ on other propositions $A$ (an axiom system)].
(2) The construction of a model. Objects and relations are exhibited which, upon suitable naming, satisfy all of the propositions $A$, and yet fail to satisfy $a$. This method has been the most successful so far invented. [...]
Klein was the first to offer a Euclidean model for non-Euclidean geometry; the objects of Euclidean geometry itself, upon an assignment of names differing from the customary one, satisfy the non-Euclidean axioms.
More than anybody else has Hilbert, through the ingenious construction of suitable arithmetical models, contributed to the clarification of the logical relations that connect the various parts of the geometrical system of axioms.
We can find the term used by Alfred Tarski in an address on the foundations of the geometry of solids given to the First Polish Mathematical Congress, held in Lwow in 1927. A summary of the address appeared in French under the title ‘Les fondements de la géométrie des corps’, 1927.
Thus the connection with "representation" is due to the technique of interpreting a geometrical system into a different one.