So that there can be no mistake, I will list the relevant definitions exactly as they appear in my edition of the Principia, which is the translation by Bernard Cohen and Anne Whitman, and provide some observations that take advantage of modern mathematical notation. (We could cite the original Latin, but it no longer common for academics to know how to read Latin anymore.) Much clarification is needed to address your question.
Definitions
- Quantity of matter is a measure of matter that arises from its density and volume jointly.
Let's call this quantity of matter "mass" and denote it by $m$. Observe that Newton defines $m := \rho V$, where of course $\rho$ is density and $V$ is volume. He does not define $\rho := m/V$, as is common in introductory physics textbooks. That is, $\rho$ and $V$ have priority for Newton as directly measurable quantities. How so? We take as self-evident that identical materials have identical densities. We can define two objects to have equal volume if they displace equal volumes of water. Then, for two objects having equal volume, we define one to be more dense than the other if the one tips a balance scale in its favor. These considerations show that Newton's definition here is inspired by Archimedes and reminds us (as the struggles of every intro physics student show) that mass is not necessarily an intuitive concept (in fact, it is often confused with weight).
- Quantity of motion is a measure of motion that arises from the velocity and the quantity of motion jointly.
We call this quantity of motion momentum and denote it as $\vec p$. You are right to observe that Newton considers momentum to be the true quantity of motion. This contrasts with Galileo and Leibniz who believed that the true quantity of motion was vis viva or "living force", which is defined as $mv^2$ or twice the kinetic energy as it is known today (cf. "virial theorem", where the mean potential energy is twice the negative of the mean kinetic energy). More interestingly, the Lagrangian and Hamiltonian formulations of classical mechanics (insofar as they are derived from the Newtonian framework) would seem to suggest that energy is the true quantity of motion. Thus, modern physics seem to side with Leibniz in more than just notation. Of course, it is fair to ask why there should be only one "true" quantity of motion. However, the question nowadays is moot since momentum and energy are unified in the modern relativistic framework via 4-vectors. Even more interestingly, the principle of relativity hearkens all the way back to Galileo, if I am not mistaken, in his work Dialogue Concerning the Two Chief World Systems.
- Inherent force of matter is the power of resisting by which every body, so far as it is able, perseveres in its state either of resting or of moving uniformly straight forward.
We call this inherent force of matter inertia. This definition is more or less the adoption of a claim from Avicenna (who was known to the Scholastics with whom Galileo frequently quarreled) that a body moving in vacuum would not stop unless acted upon. Although Newton undoubtedly had in mind Euclidean geometry, if we interpret "straight forward" to mean along its geodesic, then we have a perfect statement of general relativity. The only thing really missing is a decision whether the speed of information transmission should be finite or infinite. Unfortunately, Newton implicitly chose an infinite speed of information transmission when he elaborated his conception of absolute time, but we can hardly fault him considering what a breakthrough it was to present his laws of motion and universal law of gravitation in the first place.
- Impressed force is the action exerted on a body to change its state either of resting or of moving uniformly straight forward.
We call the impressed force impulse. Newton here defines impressed force as the cause of a change in an object's quantity of motion, i.e., momentum. It is a philosophical principle that a thing cannot cause its own change except insofar as one part of a thing acts on another part. Thus, change requires some cause, and Newton here simply singles out those causes of change in a body's quantity of motion and gives them the name force.
Newton goes on to define central forces, but that is superfluous to your question.
Laws of Motion
- Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.
Newton's first law defines inertial frames, in which his remaining laws hold.
- A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
Using modern notation $$\text{impressed force} = \int \mathrm dt \, \vec F \propto \vec p$$ taking the derivative of both sides with respect to time yields Newton's second law in the usual differential form.
Now to answer your question:
I can restate what you say at the end of your penultimate paragraph in the following manner: "If we define 1 kg as the mass of a certain bar of platinum, how can we determine the mass of an arbitrary body?" The simplest answer is not to use a spring but rather to use a balance scale, as I mentioned in my remarks on Newton's first definition. Given the 1 kg mass, we can make a 1/2 kg mass and a 1/3 kg mass and so on. (I leave it as an exercise to the reader to figure out how to make these.) Given these, we can determine the mass of an arbitrary body to an arbitrary degree of precision.
There are some misconceptions in your last paragraph concerning using a spring. The point of using a spring is that, for small enough displacement from equilibrium, the resulting force (i.e., cause of change in quantity of motion) on an attached body is approximately linear in $x$ and thus obeys Hooke's law $F=-kx$. That a device (the spring) should exert a linear force is a matter of empirical observation (look at the change in velocity, i.e., the acceleration). This linear force is related to mass via Newton's second law by definition.
There is no circularity in the logic. $ma = F = -kx$. The first equality holds by definition, the second by empirical fact. Solving for $m$ is a viable way to calculate the mass, assuming you already know $k$. How do you determine $k$? Well, that is a topic for another question, but it shouldn't be too hard to figure out ;)
Also, you might want to rework your algebra. $m \to 2m$ does not mean you have twice the acceleration.