Nowadays it seems to be popular among physics educators to present Newton's first law as a definition of inertial frames and/or a statement that such frames exist. This is clearly a modern overlay. Here is Newton's original statement of the law (Motte's translation):
Law I. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.
The text then continues:
Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motion, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.
And then the second law is stated.
There is clearly nothing about frames of reference here. In fact, the discussion is so qualitative and nonmathematical that many modern physics teachers would probably mark it wrong on an exam.
I have a small collection of old physics textbooks, and one of the more historically influential ones is Elements of Physics by Millikan and Gale, 1927. (Millikan wrote a long series of physics textbooks with various titles.) Millikan and Gale give a statement of the first law that reads like an extremely close paraphrase of the Mott translation. There is no mention of frames of reference, inertial or otherwise.
A respected and influential modern textbook, aimed at a much higher level than Millikan's book, is Kleppner and Kolenkow's 1973 Introduction to Mechanics. K&K has this:
...it is always possible to find a coordinate system with respect to which isolated bodies move uniformly. [...] Newton's first law of motion is the assertion that inertial systems exist. Newton's first law is part definition and part experimental fact. Isolated bodies move uniformly in inertial systems by virtue of the definition of an inertial system. In contrast, that inertial systems exist is a statement about the physical world. Newton's first law raises a number of questions, such as what we mean by an 'isolated body,' [...]
There is a paper on this historical/educational topic: Galili and Tseitlin, "Newton's First Law: Text, Translations, Interpretations and Physics Education," Science & Education Volume 12, Number 1, 45-73, DOI: 10.1023/A:1022632600805. I had access to it at one time, and it seemed very relevant. Unfortunately it's paywalled now. The abstract, which is not paywalled, says,
Normally, NFL is interpreted as a special case: a trivial deduction from Newton's Second Law. Some advanced textbooks replace NFL by a modernized claim, which abandons its original meaning.
Question 1: Does anyone know more about when textbooks begain to claim that the first law was a statement of the definition and/or existence of inertial frames?
There seem to be several possible interpretations of the first law:
A. Newton consciously wrote the laws of motion in the style of an axiomatic system, possibly emulating Euclid. However, this is only a matter of style. The first law is clearly a trivial deduction from the second law. Newton presented it as a separate law merely to emphasize that he was working in the framework of Galileo, not medieval scholasticism.
B. Newton's presentation of the first and second laws is logically defective, but Newton wasn't able to do any better because he lacked the notion of inertial and noninertial frames of reference. Modern textbook authors can tell Newton, "there, fixed that for you."
C. It is impossible to give a logically rigorous statement of the physics being described by the first and second laws, since gravity is a long-range force, and, as pointed out by K&K, this raises problems in defining the meaning of an isolated body. The best we can do is that in a given cosmological model, such as the Newtonian picture of an infinite and homogeneous universe full of stars, we can find some frame, such as the frame of the "fixed stars," that we want to call inertial. Other frames moving inertially relative to it are also inertial. But this is contingent on the cosmological model. That frame could later turn out to be noninertial, if, e.g., we learn that our galaxy is free-falling in an external gravitational field created by other masses.
Question 2: Is A supported by the best historical scholarship? For extra points, would anyone like to tell me that I'm an idiot for believing in A and C, or defend some other interpretation on logical or pedagogical grounds?