This is difficult to answer for a variety of reasons. If you're looking for a published source for the phrases "intended interpretation" and "standard model" with a suitable definition, many mathematical logic textbooks will do. From a historical perspective, as suggested by the phrase "_originally_ intended", one can say a lot more. However, the key players in this historical development would never use the phrase "intended interpretation" in the modern sense since the phrase entails a distinction that they weren't aware of — that there are other interpretations!

In any case, the key players Peano and Dedekind both made their intent very clear. Especially Dedekind, who wrote several essays and letters on the nature of numbers. After some searching, I just found a digitized copy of his essay [Was sind und was sollen die Zahlen?](http://www.gutenberg.org/ebooks/21016) from 1888. More famous is "The Nature and Meaning of Numbers" (from his [Essays on the Theory of Numbers](http://www.gutenberg.org/ebooks/21016)).
In his preface to [Arithmetices Principia: Nova Methodo Exposita](https://archive.org/details/arithmeticespri00peangoog), Peano states that his system is intended to derive the principles of arithmetic.
So both Peano and Dedekind made it clear that their axiomatic system was intended to describe the natural numbers.

The fact that counting numbers satisfy the Peano&ndash;Dedekind axioms is now known as [Frege's Theorem](http://plato.stanford.edu/entries/frege-theorem/#5). Although Frege first proved this using his ill-fated _Rule V_, it was later observed that that much of Frege's work didn't use the full power of _Rule V_ and thus Frege's derivation could be legitimately called a *theorem*. (See <cite authors="Richard G. Heck, Jr." mrnumber="1233925" cite="_J. Symbolic Logic_ **58** (1993), no. 2, 579--601">Richard G. Heck, Jr., [The development of arithmetic in Frege’s Grundgesetze der Arithmetik](http://dx.doi.org/10.2307/2275220), _J. Symbolic Logic_ 58 (1993), no. 2, 579&ndash;601.</cite>) So, Frege was the first to check that the Peano&ndash;Dedekind axioms did indeed describe the counting numbers.

The distinction between "intended interpretation" and "unintended interpretation" was known to Skolem around 1915 as he explains in his early critique of axiomatic set theory ("Some remarks on axiomatic set theory", found in van Heijenoort). However, it is only in the 1930's that he first managed to demonstrate the existence of non-standard models of arithmetic.