Looking for a source for Intended Interpretation

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 essays and letters on the nature of numbers. After some searching, I just found a beautiful digitized copy of his essay Was sind und was sollen die Zahlen? from 1888; you can also find "The Nature and Meaning of Numbers" in his Essays on the Theory of Numbers. In his preface to Arithmetices Principia: Nova Methodo Exposita, 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–Dedekind axioms is now known as Frege's Theorem. 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 Richard G. Heck, Jr., The development of arithmetic in Frege’s Grundgesetze der Arithmetik, J. Symbolic Logic 58 (1993), no. 2, 579–601.) So, Frege was the first to check that the Peano–Dedekind axioms did indeed describe the counting numbers.

The modern 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.

Here are quotes from three well-known sources.

Shoenfield, Mathematical Logic (1967), page 23:

We construct a model of $N$ by taking the universe to be the set of natural numbers and assigning the obvious individuals, functions, and predicates to the nonlogical symbols of $N$. This model is called the standard model of $N$, ...

(Emphasis in the original in all quotes.)

Kleene, Mathematical Logic (1967), page 200:

Since a formal system (usually) results in formalizing portions of existing informal or semiformal mathematics, its symbols, formulas, etc. will have meaning or interpretations in terms of that informal or semiformal mathematics. These meanings together we call the (intended or usual or standard) interpretation or interpretations of the formal system.

Kleene 1967 p. 207:

A we remarked in § 37, a formal system formalizing a portion of informal mathematics has an "intended" (or "usual" or "standard") interpretation. ... The informal mathematics that we aim to formalize in $N$ is elementary number theory. So for the intended interpretation, the variables range over the natural numbers $\{0, 1, 2, \ldots\}$, i.e. this set is the domain. ... The function symbol $'$ is interpreted as expressing the successor function $+1$, and $0$ ("zero"), $+$ ("plus"), $\cdot$ ("times") and $=$ (equals) have the same meanings as those symbols convey in informal mathematics.

Here Kleene explicitly speaks of the interpretation as referring to the numbers that were known informally before the axioms of $N$ were laid out.

Kaye, Models of Peano Arithmetic (1991), Chapter 1: "The standard model", p. 10:

The structure $\mathbb{N}$ (the standard model) is the $\mathcal{L}_A$ structure whose domain is the non-negative integers, $\{0, 1, 2, \ldots\}$ and where the symbols in $\mathcal{L}_A$ are given their obvious interpretation.

In contemporary practice, in formal arithmetic, it is normal practice to use the term "natural numbers" and the symbol $\mathbb{N}$ to refer to the standard natural numbers, i.e. to identify them with the informal counting numbers (e.g. this is Kaye's convention, and many others'). The need for a distinction between standard and nonstandard models is particularly evident in my own field of Reverse Mathematics; we have a different convention that $\omega$ refers to the standard numbers and $\mathbb{N}$ refers to an arbitrary model at hand (e.g. Simpson's Subsystems of Second Order Arithmetic).

Part of the issue here may be that the meaning of the term "standard model" $\mathbb{N}$ can be interpreted in several ways. From the perspective of a certain kind of realism, it refers to the "actual" counting numbers. From the point of view of a certain kind of formalism, it refers to the natural numbers in whatever metatheory is being used at the moment, so that the "standard numbers" are the ones that are metafinite and "nonstandard models" have numbers that do not correspond to numbers in the metatheory. In any case, the notation $\mathbb{N} = \{0, 1,2, \ldots\}$ is intended to convey that $\mathbb{N}$ is identified with the usual counting numbers $0$, $1$, $2$, $\ldots$ from basic arithmetic, whatever we think those are.

Here's a reference: Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, Springer, 2010. On page 42 he defines the operations $+$ and $\cdot$ on $\mathbb N$ as having their "ordinary meaning". On page 62 he says that "interpretation" = "model". Finally, on pages 105-106 he defines $\mathcal N = (\mathbb N, 0, S, +, \cdot)$ and calls $\mathcal N$ the "standard model". ($\mathbb N$ is defined as the set of natural numbers including zero on page xix.)