The author of this claim assumed that Prophet Muhammad peace be upon him had thought that the moon is further in distance to the earth than the stars to the earth. He presented Noble Verses 71:15-16 which state "See ye not how Allah has created the seven heavens one above another, and made the moon a light in their midst, and made the sun as a (Glorious) Lamp?."
What the author didn't understand from these Noble Verses is that Allah Almighty claimed that the sun is the source of light "sirajan" and the moon is the reflection of light "Nooran". The moon is not the source of light.
Now, in Noble Verse 71:16 "and made the moon a light in their midst, and made the sun as a (Glorious) Lamp?" indicates that the light whether comming from the sun (the source) or the moon has no limits. The light can travel from one heaven to the other freely.
The Noble Verses did not say that the moon belongs to the seven heavens or is further in distance to the earth than the stars to the earth. The Noble Verses clearly talked about the light that comes from the sun and from the moon traveling freely through the seven heavens "...a light in their midst... (71:16)".