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[OngBonga]

ANY constant velocity results in escape. If I'm going 10^-[big] m/s in any direction without changing speed ever, then I am not in orbit around anything.

I note you went from "velocity" to "speed". The problem with saying the Moon is accelerating is that you're observing this acceleration from Earth, not the Moon. Is the Earth accelerating as it moves along its elliptical orbit around the Sun? Well, sure, if we put the Sun in the centre of our model. But if we put the Earth at the centre, it's everything else that accelerates.

Acceleration due to gravity is not a true acceleration. It's an observed one from another frame of reference. Something that is in freefall (such as a body in orbit) is not accelerating, it is moving at a constant velocity.

[MadMojoMonkey]

I note you went from "velocity" to "speed".

Not really. I simply decoupled the direction and magnitude of the velocity to talk about them individually.
"In any direction without changing speed ever" is just a re-phrasing of "constant velocity."

The problem with saying the Moon is accelerating is that you're observing this acceleration from Earth, not the Moon. Is the Earth accelerating as it moves along its elliptical orbit around the Sun? Well, sure, if we put the Sun in the centre of our model. But if we put the Earth at the centre, it's everything else that accelerates.

Whether or not an object is accelerating is unambiguous. All observers can answer this question about a given object at a given time and all observers will agree on the answer, once relativistic effects are accounted for. In other words, absolute acceleration is a physical property.
Compare to, absolute position is not a physical property. However, relative position is. I.e. the spacetime interval between 2 objects is invariant.

If you choose a non-inertial reference frame (by saying the Earth is the center), then you can't rely on the statements of physics which begin, "In an inertial reference frame..." You need to pull out the big guns to deal with that, and let me tell you that dealing with an accelerating reference frame is not my idea of a good time.

Acceleration due to gravity is not a true acceleration. It's an observed one from another frame of reference. Something that is in freefall (such as a body in orbit) is not accelerating, it is moving at a constant velocity.

Acceleration due to gravity is indistinguishable from acceleration due to application of a force. There is no experiment which could tell you whether you're in a closed room being constantly lifted at g by a "hand of God" or simply in a closed room somewhere near the surface of the Earth.

Light travels in logically straight lines through curved space.
Massive objects do accelerate in their orbits.

All observers can measure the relative velocity of 2 bodies in orbit about each other at various times and confirm that the relative speed has changed (for non-circular orbits), indicating acceleration.

Force is invariant. Mass is invariant. Therefore F/m is invariant and F/m = a.
If any observer (after minding their p's and q's, relativistically speaking) observes acceleration, than all observers will.
For not-too-massive objects like the Earth and Moon, which have low relative velocities, the Newtonian approximation is many orders of magnitude more significant than the relativistic correction. Ergo massive bodies are shown to accelerate to some observers who have minded their p's and q's w.r.t. Einstein's Relativity, and therefore all observers will observe the same if they are also careful in their math.