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.