Does gravity CAUSE the bending of spacetime, or IS gravity the bending of spacetime?

I think the correct answer should be that what we call gravity is a fictional force which we experience due to living in an accelerated reference frame (as opposed to an inertial one). Unlike other forces, the force of gravity disappears by a coordinate change. If a person is in a falling elevator, they experience free fall, i.e. they feel like they are floating, and they would conclude there is no force of gravity acting on them. However we at the surface of the Earth would say that clearly the force of gravity is causing the elevator to plunge ever faster towards the ground.

Of course the solution to this odd state of affairs is that gravity is not a force at all. We live in a four dimensional universe with a pseudo-Riemannian geometry in which freely falling objects move along geodesics, or lines of shortest space-time distance. Because the geometry can be intrinsically curved (like the surface of a sphere), those geodesics are not what we think of as straight lines. The person inside the elevator moves along a geodesic, while we on the surface of the Earth are accelerated and do not move along a geodesic. The space-time paths (or worldines) of the elevator and the ground underneath it are not straight lines, and so they intersect at some point. That intersection is the point in space-time at which the elevator hits the ground.

One way to think of this is to consider two ants walking along lines of longitude on a globe. Lines of longitude are great circles, and are geodesics of the sphere. The two ants start at the equator on different lines of longitude both heading due north at the same speed. Their paths are initially parallel to each other, but as they move along the curved surface the distance between them shrinks until they eventually collide at the North Pole. It appears as though there is a force which is pulling them together, but in fact the force is fictitious, the reason they got closer is because on the sphere the geodesics converge and cross each other, unlike in flat space where the geodesics are straight lines which never cross. If the globe is very large, the ants will never know that they are moving on a curved surface, and so would conclude that there must be some force which attracts them. This is the fundamental picture for how "gravity" works from the perspective of General Relativity.

Now to your question, the difference is subtle. While what we refer to as "gravity" is subject to semantics, there is something more profound going on. General Relativity is usually referred to as a "theory of gravity", in which case we can think of the answer as the latter: by definition, gravity is the bending of space-time. On the other hand if we think of gravity as a force, the apparent force of gravity is essentially caused by the fact that space-time is curved. But we can essentially take this logic in circles if we think too much about it, it all depends on what we define "gravity" to be.

But deeper than this is the question of what causes gravity? In classical mechanics we are told that gravity is caused by mass, in the sense that massive bodies have a gravitational field which causes them to attract. But we know that's not the right picture. So to generalize your question, is spacetime curvature caused by mass? In some sense yes, in some sense no. Einstein's equation reads

$$G_{\mu\nu} = \kappa T_{\mu\nu}$$

where $\kappa$ is a constant, the tensor $G_{\mu\nu}$ is a function of the metric, which encodes the curvature of spacetime, and $T_{\mu\nu}$ is the stress-energy tensor which encodes the matter/energy content of the universe.

Because the theory of General Relativity is fundamentally four dimensional, and there is no preferred direction to call "time", we must essentially solve Einstein's equation "all at once". Clearly the matter content of the universe will determine the curvature of the universe, while the curvature of the universe will tell the matter how to move. So you have a sort of chicken and egg problem: matter tells space how to bend and space tells matter how to move.

There is a Hamiltonian (i.e initial value) formalism for GR which works for globally hyperbolic spacetimes (that is, it is not valid for all possible spacetimes). It is called the ADM formalism (named after Arnowitt, Deser, and Misner). It does allow one to set up initial conditions for a spacetime (initial curvature and matter/energy state) and compute the evolution of that spacetime and its matter content over "time" in a way that is generally covariant (does not violate relativity of observers). But this still does not separate the inherent link between space-time curvature and matter/energy content.

As an interesting related question, one could ask whether a massive particle moving through space can interact with itself gravitationally? That is, the mass of the particle distorts space-time and therefore alters its trajectory. There is a similar question at the end of Jackson's "Classical Electrodynamics" regarding accelerating charged particles interacting with their own radiation. I believe his conclusion is that such processes are not really considered because they would create such small corrections. In the context of GR, I would guess such questions fall in the realm of Quantum Gravity.

As to your last question, perhaps you meant "in the absence of space-time curvature". In which case the answer is no, the apple would not fall, all objects would move in straight space-time paths which never intersect and so would always remain at the same distance from each other.

In addition to P. G. A.'s answer:

Would, in the absence of spacetime, my apple still fall to the earth?

In the absence of spacetime there would be no you, no apple, no fall, no to and no earth. Spacetime is the basic framework in which the universe plays. It is the most basic known fabric of the universe.

Does gravity CAUSE the bending of spacetime?

What causes the bending of spacetime is the presence of a massive object.

IS gravity the bending of spacetime?

Yes.

Would, in the absence of spacetime, my apple still fall to the earth?

I do not understand exactly what you mean. However, what I can tell you is that, since the spacetime is curved by the presence of the Earth, the apple will follow the geodesics and therefore it will fall to the Earth.