"Massive objects like stars and planets warp the fabric of spacetime, and this curvature affects the motion of other objects, which can be understood as the force of gravity...."
this statement is incorrect, and should be the other way around:
The force of gravity can be misunderstood as a warp in spacetime and curvature of the "fabric of spacetime", which is actually an imaginary geometric model construct of humanity.....extended well beyond its applicability as a model of space and time.
"Understanding this relationship [gravity energy] is key to grasping both classical mechanics and modern physics."
Exactly!
Since all objects, inclusive EM-waves, are influenced by gravity, it is necessary to look at gravity as a result of how waves propagate in the medium, i.e., we need to understand the medium's properties of wave transportation.
Since gravitation is a phenomenon with reverse direction related to the generating particle, the medium needs a property, which allows energy to flow back to the particle's center.
Thus, the medium has a very tiny reflection factor K=8.13434(6)∙10⁻³⁴ per wave period. This constant is derived from
K = 2G∙mᵥₑᵥ²/(ħc)
with mᵥₑᵥ the mass-equivalent of the so-called Vacuum Expectation Value, the energy of the Higgs-field (246.21965(6) GeV). It is the ratio of the gravitational energy between the two VEV-masses, and the energy of a wave, where the two masses appear in the nodes of the wave, i.e. two times per wave period.
We can imagine this as any wave "excites" the vacuum and thus transports this tiny part K of its forward energy temporarily to the medium. The medium just reflects this part K and propagates it exactly backwards -> energy-backflow.
This energy-backflow already is gravitation. The forwarded wave (wavelength λₒ) gets "redshifted":
z = 2π∙K∙r/λₒ
If you set λₒ=λₕₐ=656.2nm, the wavelength of the Hydrogene Balmer alpha spectrum line, which usually is used to measure the distance of stars, as Hubble did, you can even calculate the Hubble constant
Hₒ = 2π∙c∙K/λₕₐ = 72.05 km/(sMpc)
Now, assume an oscillator, i.e., a focus, oscillating with a certain frequency: it radiates its energy evenly into its environment. Because of the energy-backflow, per period of the oscillator's propagated waves, the part K is reflected, contemporaneously and for all already propagated wave periods.
This huge amount of backward-directed energy parts is exactly as large as the currently propagated energy, thus, there is zero net energy outflow vs. inflow at any point around the focus of the oscillator. Both energy-flows generate a falling gradient 1/r, away from the focus.
Such an oscillator is a particle. It generates its own gravitational energy-backflow, which is "unmeasurable", as long as there is no other such oscillator or no other external wave.
Due to the energy gradient, any other (3D-extended) object "receives" different energy-backflows at its several distributed locations, thus, their own energy-backflows will focus slightly beside the current oscillator's focus -> displacement/acceleration towards the other oscillator -> gravity.
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"Curvature of spacetime" may be more clear with this model: any wave may be deviated by other non-parallel waves, thus, light or even particles (oscillators) will be "bent" in the "gravitational" field of other particles (oscillators).
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"potential and kinetic energy"
It should be clear that "potential" energy is not a real energy! This is the result of the imagination of a potential field around an object, which "may" act on another test object with a force. The real energy then is when this test object gets accelerated due to the force at its specific location within this field -> kinetic energy. The actual energy transport within this field can be more clear with the above explained model of gravitation and wave-interactions.
Thus, "potential" energy does not actually exist, it is just a word to describe the "possibility" for a real action, which then is actually based on a different point-to-point interaction.