I think you are right to be troubled by the phrase "beyond Earth's orbit" because it's a poor choice of words that muddies what's actually a slightly complicated situation when your looking at three bodies - in this case the Sun, the Moon, and the Earth.
So lets forget the "beyond" part for a moment because that implies a certain distance or at least a certain clearly delineated boundary. Even if you treat the Sun Earth Moon system as a 2D plane, that boundary would probably have to be drawn in four dimensions - at each point x,y there can be some velocity vectors vx, vy that will keep a spacecraft going around the Earth, while another vx, vy might put it free of the Earth to orbit the sun, and another vx, vy that will put it in at least a temporary orbit around the Moon. In the real 3D space it's a boundary in six dimensions in real space. In a CR3BP - a mathematical idealized system with perfectly circular orbits, it's fewer than six, but that's not the real world in a number of ways.
So I would propose you just ignore the literal interpretation of "beyond" as simply greater than a certain distance. Think of it as no longer orbiting just the Earth alone. Orbiting a body can be though of as simply "goes around and around" the body, and so you are right, a spacecraft orbiting the moon is also orbiting the Earth, and is also orbiting the Sun, and is also orbiting the center of the galaxy.
And there are fairly complicated shaped orbits that look like a hybrid between an Earth orbit and a Lunar orbit, or even an orbit which literally stays between the Earth and the Moon and those orbits get named after dead mathematicians, or athletic feats like (just for example) "Lyapunov" and "back-flip or Lunar Cycler" orbit.
But when a spacecraft starts circling the moon, going around and around the moon, it is really common for people to say that it is no longer orbiting the Earth. It certainly is also in orbit around the Earth, but usually people will say it's "primary" is now the Moon since that's the body it is now most closely associated with now.
Experienced "rocket scientists" know what each other mean when they say it's not in orbit around the Earth any more, and in the back of their mind they still know it is indeed orbiting the Earth.
So to answer your question: There is no simple, absolute, closed boundary in 3D (x, y, z) space where you can point and just say anything on one side is definitely in Earth orbit, and anything on the other side is definitely not in Earth orbit.
Let me know if this helps!