The word scientist was actually invented by a scientist, the British Anglican priest William Whewhell, who coincidentally also thought up phycisist, and is today remembered for little else outside of academic research.

The essay Science: The Story of a Word (Annals of Science, vol. 18, no. 2, 1962) traces the meaning of "science" and "scientist", both philosophically and linguistically, through the ages (the whole thing is worth a read):

The name scientist was first propounded in the Quarterly Review for March, 1834. The anonymous reviewer made the suggestion, too jocularly, however, to be taken entirely in earnest, in the course of a review ... From Todhunter's biography we learn that the reviewer was William Whewell (1796–1866). ...

Quoting from that article (which I haven't found the whole text for):

The tendency of the sciences has long been an increasing proclivity of separation and dismemberment. The mathematician turns away from the chemist; the chemist from the naturalist; the mathematician, left to himself, divides himself into a pure mathematician and a mixed mathematician, who soon part company; the chemist is perhaps a chemist of electro-chemistry; if so, he leaves common chemical analysis to others; between the mathematician and the chemist is to be interpolated a 'physicien' (we have no English name for him), who studies heat, moisture and the like. And thus science, even mere physical science, loses all trace of unity. A curious illustration of this result may be observed in the want of any name by which we can designate the students of the knowledge of the material world collectively. We are informed that this difficulty was felt very oppressively by the members of the British Association for the Advancement of Science, at their meetings at York, Oxford and Cambridge, in the last three summers. There was no general term by which these gentlemen could describe themselves with reference to their pursuits. Philosophers was felt to be too wide and lofty a term, and was very properly forbidden them by Mr. Coleridge, both in his capacity of philologer and metaphysician; savans was rather assuming, besides being French instead of English; some ingenious gentleman [this refers to Whewell himself] proposed that, by analogy with artist, they might form scientist, and added that there could be no scruple in making free with this termination when we have such words as sciolist, economist, and atheist — but this was not generally palatable; others attempted to translate the term by which the members of similar associations in Germany have described themselves, but it was not found easy to discover an English equivalent for Natur-Forscher. The process of examination which it implies might suggest such undignified compounds as nature-poker, or nature-peeper; but these were indignantly rejected.

He later suggested the new word more formally a work under his own name, "The Philosophy of the Inductive Sciences, Founded Upon Their History" (London: John W. Parker, 1840).

The transition from "natural philsopher" into the more specific words we know today seems to have taken more than half a century, with some scientists (such as Lord Kelvin and Michael Faraday) initially demurring, and during which random attempts at getting savant and 'scient established failed. By the time 1911 edition of The Concise Concise Oxford Dictionary of Current English came out — which was the first published version of the OED that has all of A-Z (the first full dictionary only came out in 1928) — scientist was in, and entries described people as "scientists".

What we today call "science" was for most of its history called "natural philosophy," which is to say, the branch of philosophy that approached questions of the natural world. Today of course we consider this a funny anachronism, but it makes perfect sense if you look at the origins of "science" in people like Aristotle, whose work was more "natural philosophy" than "science" — it really was a form of philosophy, for better or worse, but it described things like the elements and the body and animals and what have you, with absolutely zero interest in empiricism or experimentation or other hallmarks of "science."

As noted elsewhere, "scientist" and many other "-ist" terms did not arise until the 19th century. What is of note here separate from the etymology is that this is exactly the time that what we call science was "professionalizing" in England. That is, it was changing from "speculations that rich people do or sponsor in their spare time" and into "a job that people could go through a rather regular career path to end up doing." The divisions between scientific disciplines start to become more iron-clad as specialization takes root. Universities becomes more and more the sites of scientific research (as opposed to finishing school for the upper class). And so on. This process took about a century; by the end of the 19th century you still have some people who are sort of in the original model (like Darwin, who straddles the period), but you also have people who are very much recognizable as "modern" scientists in their approach to the profession (like Huxley and Wallace).

As for Newton — one might even want to call him a natural philosopher even if one did want to use modern terminology. He is very much of a previous period, and his intellectual interests were much broader than what we would typically call a scientist today. Modern scientists would reject the majority of Newton's interests as being non-scientific — alchemy and theology being chief among them. That's no sin against Newton, to be sure, but I just point it out to emphasize that seeing the part of Newton's work that concerns the natural world as a form of philosophy is not so alien if you look at the work itself in its own context, rather than the very refined and selective versions of it that you would today encounter in a physics or astronomy course (much of which is Newton-translated-into-modern-science by people like Laplace). Newton himself believed that his equations were of great philosophical import and purpose — in his "system of the world" you saw not only equations governing the rotation of the planets, but the hand of God himself, keeping everything from going awry. (Laplace showed that they would not go awry even without a hand of God — he stamped God out of Newton's equations, so to speak.)