The concept of inertia is indeed useful in two ways. I think your notion of it as a technical promotion of the everyday word "sloth" (without the baggage given it by the Roman Catholic translation of the "deadly sin" Ἀκηδία) as extremely close to the mark. In physics the notion of "inertia" has two, very alike uses:
The first is practical, through a weak form of D'Alembert's principle. The notion arises where we look at a system from an accelerated frame of reference and treat it as a non-accelerated one: to keep the things making up the system "together" and "still" relative to the accelerated frame, we imagine that each of the system components are exerting a force of inertia (in the sense exactly described by Newton in your quote) on the system that tries to "tear it away" from the frame of reference wherein our discourse takes place. This "force" arises from each component's "sloth", i.e. resistance to any change of their state of motion from a non accelerated frame (let's leave aside the term "inertial frame" for this latter notion for now). There has to be something tethering each of the system components to the accelerated frame to resist the "force of inertia" that each of the components exerts in "trying to tear away" from the frame and resume a uniform state of motion. Thus, in designing a centrifugal pump in this way, we would imagine the impellor sitting still, but each of the blades exerts its centrifugal force on the impellor's hub and we thus see that the hub and blades are in a state of tension to resist this centrifugal force and must accordingly be designed so that they may be strong enough to yield this resistance. From a non-accelerated frame, we would simply see the blades making circular paths, and thus we conclude that they are accelerating, so, by Newton II, we know that the hub must be pulling the blades radially, i.e. providing the centripetal force needed to set up this accelerated motion. Sometimes D'Alembert's Principle is thought of simply a re-arrangement of Newton's second law, and thus derivable from the latter, but this is not so as discussed in QMechanic's answer here to the Physics SE question "Deriving D'Alembert's Principle". Moreover, it is indispensable in rotating rigid body problems. In this kind of problem, if we try to work only in non-accelerated frames, Euler's Second Law of Motion becomes highly awkward, because the inertia tensor $I$ of a spinning body is constantly changing relative to a non-rotating frame. It is much easier to fix our frame to the spinning body, thus benefitting from a constant inertia tensor $I$ and live with the inertial forces $\omega\times(I\,\omega)$ in the Euler equations $M = I\, {\rm d}_t \omega + \omega\times(I\,\omega)$
The great theoretical utility of the notion of inertia is as inertial mass: this notion is useful simply by differing from the notion of gravitational mass. Without a clear understanding of the stark differences between these two notions, there could be no discussion of the Equivalence Principle (see Wikipedia page of this name). In this form, the notion of inertia is an indispensable part of the epistemology of the General Theory of Relativity, so I shall now concentrate on talking about this use of the "inertia" notion.
Inertia, Gravitational Coupling and The Equivalence Principle
So now we look at the meaning of the word "mass": it actually has two (and possibly three) in principle distinct meanings:
As "inertia" or "inertial mass" it is a measure of the body's "sloth" or "shove resistance" as discussed above, i.e. inversely proportional to its acceleration under a unit imbalanced force (i.e. inversely proportional to the body's "response" to a standard force). Thus this notion is expressed by the quantity $m_I$ in Newton's second law $\vec{F} = m_I \,\vec{a}$;
As a "coupling constant" describing how strongly a body is influenced by a gravitational field, i.e. how much nett force a unit gravitational field imparts to a body (in the Newtonian notion of gravity). Thus this notion is expressed by the quantity $m_g$ when we say that a small test mass of gravitational mass $m_g$ in a gravitational field $\vec{g}$ feels a force $m_g\,\vec{g}$;
A third possible notion, not really important here, is as a measure of a particle's "stay puttability" as I discuss in my answer to the Physics SE question "Can mass be directly measured without measuring its weight?". This is simply how still a body of a given, standard degree of localisation in space can be made and still conform with the Heisenberg Uncertainty Principle.
Ponder the first two carefully and take heed how much in principle different they are as notions. Without further information, experimental results or postulates, I hope you will agree that there is no possible way whereby these two notions could a priori proven or even guessed to be the same.
The weak equivalence principle asserts that notions 1) and 2) above are the same and, for any body, regardless of its makeup or quantum state, we have $m_I = m_g$ (see my footnote 3). These two are not just alike. They are exactly the same. This is a stunning assertion and it still dazzles me to this day, even though I am fifty years old and first read about it when I was fourteen (I did, however, take a further six years to fully appreciate its significance).
Given their vast conceptual difference as physical notions, any assertion that they are the same must encode real, falsifiable physics about gravitation. For what it means is that any small mass, regardless of makeup or quantum state, with a given initial velocity in a gravitational field must undergo exactly the same motion. This principle has been clearly recognised by many scientists for nearly fifteen hundred years. In the sixth century CE, John Philoponus (see Wiki page of same name) said of experiments potentially falsifying the equivalence principle:
"But this [view of Aristotle that the time taken for a body to fall a given distance is inversely proportional to its weight] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small."
Galileo certainly knew the equivalence principle and his famous experiment dropping different weight balls from the Tower of Pisa was almost certainly, in reality, done in about 1586 by Simon Stevin dropping balls from the Delft churchtower (see discussion in Equivalence Principle Wiki Page).
Many careful experiments probing the equivalence principle's correctness have been done; amongst the most famous are the Eötvös Experiment (see Wikipedia page of same name) as well as those of Newton (in finding that pendulums of the same length have the same period, independent of length) and by the commander of Apollo 15, David Scott, when he dropped a feather and a hammer from the same height on the Moon to see them both hit the ground at exactly the same time.
So now on to General Theory of Relativity. Einstein was convinced that the Equivalence Principle would lead him to his GTR and from very early on in the piece he kept returning to this principle. The principle shows itself very blatantly in his early works before the full GTR paper of 1916. In:
A. Einstein, "Über den Einfluss der Schwerkraft auf die Ausbreiitung des Lichtes", Annalen der Physik, 35, 1911 English version "On the Influence of Gravitation on the Propagation of Light" is here)
he uses nothing but the equivalence principle very directly and on its own to derive, by very simple and clear arguments, some of the important, readily falsifiable results that would follow from his later 1916 paper:
A. Einstein, "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49, 1916 (English translation, "The Foundation of the General Theory of Relativity" is here)
In the latter paper, indeed in the teaching of GTR, the equivalence principle seems to slip into the background a little (a great deal, in some modern texts) and often the direct assertion of the equivalence of mass notions is overshadowed in modern texts by something like the following statement:
The tangent space to the spacetime manifold solution to the Einstein field equations is Minkowskian
or
Spacetime is locally Minkowskian
or something like this. This is indeed a reasonable, indeed a stronger, statement of the equivalence principle, but it does, in my opinion, need some further explanation. It's at first glance quite different from the Einstein version of the equivalence principle:
The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime
The way that the equivalence principle is encoded is, in my opinion, some of the reason why inertia is not discussed very much in relativity. The EP is actually part of the building materials for the GTR: the very assertion that spacetime under the influence of "matter" (anything with energy content and thus gravitational mass) is a differentiable, indeed, pseudo-Riemannian manifold fully encodes the EP. So, the mere choice of the geometrical object, before we even contemplate writing down the Einstein Field Equations, or indeed the physics behind them, fully encodes the EP. A manifold is locally like a Euclidean (or, in GTR, flat Minkowskian) space: there are other geometrical objects, notably an Algebraic Variety that we might have chosen to describe the "curvature" of spacetime with and which are more general than manifolds and which do NOT encode the EP. To examine the manifold and why it encodes the EP, I'm going to give my version of the Einstein equivalence principle:
For any chosen, positive precision $\epsilon>0$, there is a magnification $M$ such that if you look at the spacetime manifold with this high enough magnification, you will see a laboratory indistinguishable (to within the chosen precision) from the main cabin of Salviati's Ship
Salviati's ship (see the Wikipedia page for "Galileo's Ship") was, of course, a thought experiment wherein Galileo asserts the impossibility of telling whether or not a ship is moving uniformly by any experiment which does not look to an outside reference. The mere fact that the spacetime manifold has a Tangent Space at every point alone means that if we zoom into the manifold enough so that our little ship takes up a small enough volume in space time, then there exists a frame of reference, that moving along a geodetic line, such that if the ship were stationary in that frame of reference, the Salviati thought experiment would hold. It might (inside a highly curved region like near a black hole) have to be a volume that it is comparable in size to an atomic nucleus, but that's OK in GTR: GTR is a classical theory that does not see the World's granularity: in principle there is always a Salviati ship, even if it is the size of a proton and exists only for $10^{-20}$ seconds. This is a freefall frame, the most general concept of an inertial frame, or a "slothful frame" that describes a (geodesic) flow on spacetime arising in the absence of external forces, and that any accelerated motion relative to that frame requires an unbalanced force. It describes how something moving through spacetime "wants to move" and has a "stubbornness to move thus" and must be compelled by unbalanced force to move differently.
So suppose we are on Salviati's ship, freefalling in a uniform gravitational field and the equivalence principle did not hold, and no frame of reference were Minkowskian (an inertial frame, in Special Relativity). The butterflies, being of a different makeup, might accelerate differently from the waterdrops from the bottle, and the ship's cat, being of very different makeup, would have accelerated relative to the scene and been lost long ago! The whole scene can only stay fixed, with all its constituents staying at the same relative positions, if the equivalence principle holds. Thus we see that the differentiable manifold conception of spacetime can hold only if the equivalence principle is true.
So there is always a local "inertial frame" (I actually like the word "freefall frame" better) in General Relativity. We have come the full circle: for now how do we describe your frame of reference as you sit "stationary" on the Earth's surface reading this? Think about your bottom: you feel it is being pressed by your seat. You conclude that there must be a force pressing you into the seat: our mother tongue has a word for this force: your weight. But this is an inertial force in the sense described at the very beginning of my answer. For it so happens that, in the presence of the Earth, the true inertial frame, the true cabin of the Salviati Ship, as described by GTR, is one beginning stationary relative to you but which is "accelerating" relative to you at $g {\rm m\,s^{-2}}$ towards the centre of the Earth. The matter of your seat must therefore push against you with force $m\,g$ upwards to beget your acceleration relative to the Salviati ship. But, of course, we find it easiest to think in a frame of reference that is stationary relative to our Earthly home. So, in this accelerated frame, we feel the inertia of our bodies as they "try to tear away" and follow their natural, inertial frames.
See Eduardo Guerras Valera's wonderful answer to "How (or why) equivalence principle led to Einstein field equations?" for a fuller description of how Einstein seems to have embedded the EP into the GTR - the modern manifold concept I described was not how people thought about manifolds in Einstein's day, when they thought of them as needfully being curved objects in a higher dimensional Euclidean space. The two conceptions were only shown to be equivalent notions in the 1940s by Hassler Whitney and 1950s by John Nash (the mathematician depicted by Russell Crowe in the film "A Beautiful Mind").
Some theorists believe that the EP is so NONtrivial that its very breakdown (actual experimental falsification) may be the first place where we see in practice the General Theory of Relativity yield to more general, yet to be developed quantum theory of gravity. See the discussion in vnb's answer to the Physics SE question "Does quantum mechanics violate the equivalence principle?". Indeed Paul Davies in his article "Quantum mechanics and the equivalence principle" shows a possible chink: for quantum particles tunnelling to regions in a gravitational field whence they are classical forbidden, the tunnelling depth depends on the particle mass. Also, the problem of whether an electrical charge on the Earth's surface radiates does not seem to be fully resolved. See Ben Crowell's answer to the Physics SE question "Does a constantly accelerating charged particle emit em radiation or not?".
Strictly speaking, all the physics of the Equivalence Principle would be encoded by an assertion that $m_I = \lambda\,m_g$, where $\lambda$ is any constant, so $m_I=m_g$ encodes the EP together with a choice of scaling constant. We choose to define the universal gravitation constant $G$ in Newton's law of gravitation so that $\lambda = 1$. Actually, in General Relativity, in some ways it would make more sense to define $G/(8\pi)$ to be the gravitation constant, in which case the equivalence principle would be $m_I = \sqrt{8\,\pi}\,m_g$.
