1. In Newtonian physics, when you are talking about kinetic energy and potential energy of a system, say $L_p=\dot x^2/2 -V(x)$, you refer to a kinetic energy $T=\dot x^2/2$ and a potential energy $V(x)$ of a particle. Even if the potential is generated by some kind of fields $L=L_p+L_F$, the only thing you need to care is the particle part $L_p$ (the equation of particle), in order to define the kinetic energy and potential energy.
2. For general relativity, let us first define a kinetic energy for a special case, i.e. a free falling particle, whose Lagrangian is $2\mathcal{L}=g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\mu}{d\tau}$. From my side, the kinetic energy is obvious, it is $\mathcal{L}$. If you can not see what I see, you can write the geodesic equation, then you will find the first integral of motion, in other words, $\mathcal{L}$ is a constant along the geodesic, which can be regarded as "kinetic energy" of this particle. Meanwhile it is consistent with Newtonian limit up to a constant.
3. Now let us turn to a more general case, where there is a "potential" generated by a kind of field, say EM. First, the "kinetic energy" would not change; Second, the only thing new is Lorentz force, which can be cast in the form of potential (this does not bother me, we can simply cast in a formal integral). 499
If you write down a geodesics equation, the demonstration in this item is not hard to be understood.
4. However the definitions "difference of Hamiltonian with Lagrange as 2 times potential energy, and the difference of Hamiltonian with potential energy as kinetic energy" are more challenging. For free falling particle, $\mathcal{L}=\mathcal{H}$, thus there is only kinetic energy; for the case in item 3, this definition is also consistent with above one.
5. For definitions of the kinetic and potential energies of classical fields, the situation is similar.
6. In <PERSON>'s general relativity, gravity is a background geometry, rather than a force/potential, thus there is no need to separate the particle with its background. Therefore the kinetic and potential energies in item 4 refer to the particles with non-gravitational force.
Edit: 7. How to separate the "Gravitational potential" in the free falling particle case in the sense of Newtonian physics?
I think a lot of people did this problem in his beginning course of GR. All we need is two approximations: 1, slow moving condition and 2, weak field condition. Based on these two conditions, the potential energy can be formally extracted from the equation of motion, which is relevant to the 0-0 component of perturbed metric.
Edit: 8. In fact, if you get rid of energy-momentum tensor, you even can define "kinetic" and "potential" energies for gravity via its Hamiltonian constraint. 651