They mean outside of low Earth orbit (which basically means further away than the ISS). The phrasing is not ideal.
> Earth's gravitational dominance extends 4x the distance to the moon
"Earth's gravitational dominance" is not a single thing; it depends on what kind of "dominance" you're talking about.
For example, even though the Moon is usually described as being in orbit about the Earth, its orbit is always concave towards the Sun. In other words, its net gravitational acceleration is always towards the Sun--even when the Earth is on the other side of it from the Sun. So by this criterion it's not in orbit about the Earth, it's in orbit about the Sun, doing a complicated do-si-do with the Earth, also in orbit about the Sun.
I'm not sure what definition of "dominance" you're using that extends the Earth's "dominance" to 4 times the distance of the Moon.
It’s about the suns gravitational pull on the moon dominating over the Earths gravitational pull on the moon, but that due to the centrifugal force (there isn’t one, so conservation of angular momentum) the Earth's gravitational pull dominates.
The claim about centrifugal force refers to the Hill sphere, which is a different notion of "gravitational dominance". The basic idea behind that is that, while the Sun's force on the Moon is greater than the Earth's, it varies in space, in the region where the Earth and Moon are orbiting, much less than the Earth's does. So we can "subtract out" the Sun's gravitational force, so to speak, since we can approximate it as constant in the region we're interested in.
The video, however, bungles this somewhat, because its claim about "centrifugal force" is made in a frame which is centered on the Sun--but rotating at the same rate the Earth revolves around the Sun. But nobody actually uses such a frame! Doing that would be silly. The natural frame for us on Earth to use if we "subtract out" the Sun's gravitational force to analyze the Earth-Moon motion is a frame centered on the Earth.
In this frame, we can say that the Moon orbits the Earth, not because there is some "centrifugal force" canceling out the Sun's force, but because we've subtracted out the Sun's force by centering our frame on the Earth. Or, to put it another way, we're treating the whole Earth-Moon system as freely falling in the Sun's gravitational field, and as long as the Sun's field is, to a good enough approximation, constant in the region we're interested in, we can simply ignore the Sun's gravitational force. (This viewpoint is much more natural in General Relativity, where "gravity" is not a force at all to begin with.) Such a frame is called an "Earth-Centered Inertial" frame, and it's the frame that's being used, for example, to manage the Artemis II spaceflight.