The "classical" references for the sources and evolution of Newton's mechanics are : Alexandre Koyré, Richard S.Westfall and I.Bernard Cohen.
Ivor Grattan-Guinness' comment is [in the Norton edition of the book] at page 259 refers to Richard S. Westfall, Force in Newton's physics: the science of dynamics in the seventeenth century (1971), Ch.8.
I've no acces to it, but we can see a detailed discussion of Newton's drafts of Principia into :
I'll try to sumamrize his argument :
De motu started with two definitions and two hypotheses. Definition 1 drew upon the lesson about circular motion that Hooke had taught him in 1679 to introduce a new word into the vocabulary of mechanics.
I call that by which a body is impelled or attracted toward some point which is regarded as a center centripetal force.
[...] The legacy of De gravitatione appeared in Definition 2, about rectilinear motion.
And [I call] that by which it endeavors to persevere in its motion in a right line the force of a body or the force inherent in a body.
Hypothesis 2 extended the definition into a general conception of motion.
By its inherent force alone, every body proceeds uniformly in a right line to infinity unless something extrinsic hinders it.
[...] The tract attempted to derive orbital motion from the interaction of two forces: inherent force, which maintains rectilinear motion, and centripetal force, which continually diverts it. To compound the two, Newton employed the parallelogram of forces, which he inserted later as Hypothesis 3, although he made use of it before he added its formal statement.
When it is acted upon by [two] forces simultaneously, a body is carried in a given time to that place to which it would be carried by the forces acting separately in succession during equal times. 
Three versions of De motu survive. The second version, late 1684, which is written in Halley's hand and is identical to the copy in the register of the Royal Society, was merely a fair copy of the first with small additions (such as Hypothesis 4) which Newton had indicated his intention to insert. The third version, on the other hand, contained the beginning of Newton's reconstruction of his dynamics. 
In the third version, early 1685, Newton moved Hypotheses 3 and 4, the parallelogram of force and the measure of distance described under a centripetal force, to the status of lemmas. To the remaining two hypotheses modestly altered, he added three more and then rejected the name "Hypothesis" in favor of a new one,
"Lex." Hence Newton's laws of motion initially numbered five [emphasis added]. From its new position at the head of the list, Law 1 continued to
assert that a body moves uniformly by its inherent force alone. In Law 2, Newton attempted to define the action of force.
The change of motion is proportional to the impressed force and is made in the direction of the right line in which that force is impressed. 
Law 3 asserted that the motions among themselves of bodies included in a given space are the same whether the space is at rest or moves uniformly in a straight line without circular motion. Law 4 asserted that the mutual actions of bodies do not change the state of motion or of rest of their common center of gravity. The first draws on the principle of inertia; the second extends it to systems of bodies considered as unities. Newton later demoted the two to the rank of corollaries to the laws of motion, but both made their way into the Principia. Law 5, a restatement of Hypothesis 1 of the first version, contained a supposedly empirical statement about the resistance of media. 
Added August 2015
Grattan-Guinness's comments refers to Westfall's book on Force in Newton's physics, where the discussion of Newton's 1684 manuscripts on De Motu takes place in page 432-on.
Page 432 :
As far as his [Newton's] dynamics is concerned, [...] this development appears to have been completed during roughly the six months following the redaction of the original version of De motu.
In the original version of De motu, Newton derived the law of areas, the connection of the inverse square relation with elliptical orbits, and basic propositions concerning motion through a uniformly resisting medium from a dynamics less satisfactory than that he had employed in 1679, a dynamics inconsistent with itself and utterly inadequate to the load of demonstration it was asked to carry.
Page 433 :
In the MS. in which it has survived, the first version of De motu contains two stages, an original form and a set of emendations and additions. The original form started with the two definitions cited above and two hypotheses. Hypothesis one said that bodies are not impeded by a medium or by other external causes which would prevent them from exactly obeying the inherent and centripetal forces. [...] he inserted a third definition - of resistance as the force of a medium uniformly impeding motion - and altered hypothesis one to say that resistance is assumed to be zero in the first nine propositions and proportional to the velocity of the body and the density of the medium in the remaining ones.
Sometime after the original redaction, and probably at the same time as the additions mentioned above, he entered a third hypothesis in the margin and the heading ‘Hyp 4,’ although he did not write in the fourth hypothesis itself. The third one stated explicitly the dynamics already employed in the following propositions.
Hyp. 3. When it is acted upon by [two] forces simultaneously, a body is carried in a given time to that place to which it would be carried by the forces acting separately in succession during equal times.
Page 439 :
As far as Newton’s ultimate dynamics was concerned, the central problem that the first version of De motu left for solution was the resolution of the contradiction posed by his acceptance of two radically different concepts of force, inherent force which maintains uniform motion, and centripetal force which alters it. To the solution of the problem, version two, which was little more than a fair copy of the amended form of version one, added nothing. Version three, in contrast, marked a further stage in Newton’s thought [emphasis added]. Hypotheses 3 and 4 of the earlier versions were moved to the status of Lemmas, enhanced by demonstrations, but not otherwise altered. To the remaining two hypotheses Newton added three more, decided that the word ‘Hypothesis’ did not express his meaning, struck it out in all five cases, and replaced it with a new word, ‘Lex.’
In their original incarnation, then, the laws of motion numbered five [emphasis added]. The first remained substantially unchanged from version one, still asserting that by its inherent force alone a body proceeds with a uniform motion in a right line. Law two, however, underwent a significant change. The word ‘impressed’ replaced the word ‘centripetal,’ not only increasing the generality of the law, but also expressing the dichotomy that Newton was seeking to exploit. The inherent or internal force of a body maintains it in uniform motion. The force external to a body, impressed on it from without, alters its uniform motion.
The change of motion is proportional to the impressed force and is made in the direction of the right line in which that force is impressed.
With the exception of one word (the adjective ‘motive’ modifying impressed force), the law is identical to that which appeared in the Principia.
Of the remaining three hypotheses - or laws - added in this version, the last of them formalised what he had discovered in version one about the resistance of a medium. It asserted that the resistance of a meditun is proportional to its density, to the surface of the body moved, and to the velocity, all three taken conjointly. This peculiar ‘law’ experienced a peculiar fate. In the next revision, it suffered the indignity of an additional note in which Newton confessed that he did not affirm the law to be exact; it was sufficient that it be approximate. Whereupon he recognised the absurdity of what he was saying, and with a stroke of the pen removed it forever, as it deserved, from the laws of motion.
The other two laws of version three were less anomalous but in the end more troublesome. One asserted that the ‘relative motions of bodies contained in a given space are the same whether the space in question rests or moves perpetually and uniformly in a straight line without circular motion.’ The other complemented it by affirming that the common centre of gravity of a system of bodies does not change its state of motion or rest because of the mutual actions of the bodies on each other.
In a word, these two laws returned to the insights of the Waste Book and its realisation that in an inertial system two bodies isolated from outside influences can be considered as one concentrated at their common centre of gravity.
For more details, see :
From page 304-on there is the text with translation of MSS.Xa of the Cambridge U.L., named : De Motu Corportim in medijs regulariter cedentibus and of MS.Xb, named De Motu Corporum. The last one includes Definitions only, while the first one has six Leges Motus :
It is also probable that they succeeded Version III of the tract de Motu.
Here are the six Leges Motus of MS.Xa [Herivel transl.] :
Law 1. By reason of its innate force every body preserves in its state of rest or of moving uniformly in a straight line unless in so far as it is obliged to change its state by forces impressed on it. [...]
Law 2. The change of motion is proportional to the force impressed and takes place along the straight line in which the force is impressed. [...]
Law 3. As much as any body acts on another so much does it experience in reaction. [...]
Law 4. The relative motion of bodies enclosed in a given space is the same whether that space rests absolutely or moves perpetually and uniformly in a straight line without circular motion. [...]
Law 5. The common centre of gravity of [a number of] bodies does not change its state of rest or motion by reason of the mutual actions of the bodies. This law and the two above mutually confirm each other.
Law 6. The resistance of a medium is jointly proportional to the density of that medium, the area of the moved spherical body and the velocity. I do not assert this law to be exact. It suffices that it should be approximately true.