

I think the diagram is mean to be a perspective view from above the plane of the Earth's orbit, so we are seeing the moon on the far side of the Earth, rather than 'above' it.
L2 'orbits': the term 'orbit' is slightly misleading, as we tend to understand it as meaning orbiting about a large mass such as the sun or a planet. Here's my armwaving explanation of L2.
 If a probe is put in an orbit around the sun slightly outside Earth's orbit, then it will take longer than Earth to orbit the sun, and so would normally tend to lag behind Earth.
 But Earth's gravity will add to the sun's gravity, and in effect strengthen it. As such the probe has to orbit a bit faster to counteract the extra gravitational pull.
 Get the distance right, and the extra speed around the orbit exactly makes up for the orbit being bigger (and thus slower) than Earth's, so the probe still takes exactly one year to go around the sun, hence keeping pace with Earth.
 This still works if the probe is not exactly on the sunEarth line. But if it is off a little, then what it ends up doing is following a path that from the perspective of Earth is a circle around the 'L2' point. It's not orbiting that point  it's orbiting the sun  but from Earth it is going in a circle around it, so 'orbit' is the term that tends to get used.
The threebody problem, and the special restricted solutions to it, is a very challenging and nonintuitive aspect of orbital mechanics. L1, L2 are reasonably easy to understand when you look at it in the way I've described. (For L1, the point inwards of Earth, just run the argument the other way, with the Earth's pull reducing the sun's apparent gravity). L3, on the far side of the sun, is rather weirder. L4 and L5, 60 degrees ahead of and behind the Earth but in the same orbit, are just bizarre, but fall out of the maths  and turn out to be the most stable points of all.
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